Introduction
Derivation
The Rules
Examples
The Product and Quotient Rules
Bernd Schröder
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Why Do we Need the Product Rule and the
Quotient Rule?
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Why Do we Need the Product Rule and the
Quotient Rule?
Products like
ex x2 + 2 cannot be multiplied out like, say,
x3 x2 + 2 .
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Why Do we Need the Product Rule and the
Quotient Rule?
Products like
ex x2 + 2 cannot be multiplied out like, say,
x3 x2 + 2 .
2x2 + 3
cannot be converted into
x2 − 4
functions that do not involve quotients.
Similarly, quotients like
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful.
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3
Bernd Schröder
The Product and Quotient Rules
k(x) := x2 + 2
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3
k(x) := x2 + 2
gk(x) = x3 x2 + 2
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3
k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3
k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk)0 (x) = 5x4 + 6x2
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3
k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk)0 (x) = 5x4 + 6x2
6= g0 (x)k0 (x)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3
k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk)0 (x) = 5x4 + 6x2
6= g0 (x)k0 (x) = 3x2
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3
k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk)0 (x) = 5x4 + 6x2
6= g0 (x)k0 (x) = 3x2 · 2x
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3
k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk)0 (x) = 5x4 + 6x2
6= g0 (x)k0 (x) = 3x2 · 2x = 6x3
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Deriving the Product Rule.
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
h→0
h
(gk)0 (x) = lim
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
h→0
h
g(x + h)k(x + h) − g(x)k(x)
= lim
h→0
h
(gk)0 (x) = lim
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
h→0
h
g(x + h)k(x + h) − g(x)k(x)
= lim
h→0
h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
= lim
h→0
h
(gk)0 (x) = lim
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
h→0
h
g(x + h)k(x + h) − g(x)k(x)
= lim
h→0
h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
= lim
h→0
h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
= lim
h→0
h
(gk)0 (x) = lim
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
h→0
h
g(x + h)k(x + h) − g(x)k(x)
lim
h→0
h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
lim
h→0
h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
lim
h→0
h
g(x + h) − g(x)
k(x + h) − k(x)
lim
k(x + h) +
g(x)
h→0
h
h
(gk)0 (x) = lim
=
=
=
=
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
h→0
h
g(x + h)k(x + h) − g(x)k(x)
lim
h→0
h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
lim
h→0
h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
lim
h→0
h
g(x + h) − g(x)
k(x + h) − k(x)
lim
k(x + h) +
g(x)
h→0
h
h
g(x + h) − g(x)
k(x + h) − k(x)
lim
lim k(x + h) + lim
lim g(x)
h→0
h→0
h→0
h→0
h
h
(gk)0 (x) = lim
=
=
=
=
=
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
h→0
h
g(x + h)k(x + h) − g(x)k(x)
lim
h→0
h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
lim
h→0
h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
lim
h→0
h
g(x + h) − g(x)
k(x + h) − k(x)
lim
k(x + h) +
g(x)
h→0
h
h
g(x + h) − g(x)
k(x + h) − k(x)
lim
lim k(x + h) + lim
lim g(x)
h→0
h→0
h→0
h→0
h
h
g0 (x)k(x) + k0 (x)g(x)
(gk)0 (x) = lim
=
=
=
=
=
=
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
h→0
h
g(x + h)k(x + h) − g(x)k(x)
lim
h→0
h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
lim
h→0
h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
lim
h→0
h
g(x + h) − g(x)
k(x + h) − k(x)
lim
k(x + h) +
g(x)
h→0
h
h
g(x + h) − g(x)
k(x + h) − k(x)
lim
lim k(x + h) + lim
lim g(x)
h→0
h→0
h→0
h→0
h
h
g0 (x)k(x) + k0 (x)g(x)
(gk)0 (x) = lim
=
=
=
=
=
=
The derivation of the quotient rule is similar.
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Theorem.
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Theorem. The product rule and the quotient rule.
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x
k
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p0 (x) = (gk)0 (x) = g0 (x)k(x) + k0 (x)g(x),
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p0 (x) = (gk)0 (x) = g0 (x)k(x) + k0 (x)g(x),
and if k(x) 6= 0, then
g 0
g0 (x)k(x) − k0 (x)g(x)
q0 (x) =
(x) =
.
k
k2 (x)
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p0 (x) = (gk)0 (x) = g0 (x)k(x) + k0 (x)g(x),
and if k(x) 6= 0, then
g 0
g0 (x)k(x) − k0 (x)g(x)
q0 (x) =
(x) =
.
k
k2 (x)
Remember the product rule in the same order as the numerator
of the quotient rule.
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p0 (x) = (gk)0 (x) = g0 (x)k(x) + k0 (x)g(x),
and if k(x) 6= 0, then
g 0
g0 (x)k(x) − k0 (x)g(x)
q0 (x) =
(x) =
.
k
k2 (x)
Remember the product rule in the same order as the numerator
of the quotient rule.
Product The derivative of a product is the derivative of the first factor times
Rule.
the second factor plus the derivative of the second factor times the
first factor.
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p0 (x) = (gk)0 (x) = g0 (x)k(x) + k0 (x)g(x),
and if k(x) 6= 0, then
g 0
g0 (x)k(x) − k0 (x)g(x)
q0 (x) =
(x) =
.
k
k2 (x)
Remember the product rule in the same order as the numerator
of the quotient rule.
Product The derivative of a product is the derivative of the first factor times
Rule.
the second factor plus the derivative of the second factor times the
first factor.
Quotient The derivative of a quotient is the derivative of the numerator times
Rule.
the denominator minus the derivative of the denominator times the
numerator (as a quantity) divided by the square of the denominator.
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
Bernd Schröder
The Product and Quotient Rules
2
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
f 0 (x)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
f 0 (x) = (ex )0
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
f 0 (x) = (ex )0 x2 + 2
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
0
f 0 (x) = (ex )0 x2 + 2 + x2 + 2
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
0
f 0 (x) = (ex )0 x2 + 2 + x2 + 2 ex
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
0
f 0 (x) = (ex )0 x2 + 2 + x2 + 2 ex
= ex
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
0
f 0 (x) = (ex )0 x2 + 2 + x2 + 2 ex
= ex x2 + 2
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
0
f 0 (x) = (ex )0 x2 + 2 + x2 + 2 ex
= ex x2 + 2 + 2x
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
0
f 0 (x) = (ex )0 x2 + 2 + x2 + 2 ex
= ex x2 + 2 + 2xex
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Compute the derivative of f (x) = e x + 2
x
2
0
f 0 (x) = (ex )0 x2 + 2 + x2 + 2 ex
= ex x2 + 2 + 2xex
x
2
= e x + 2x + 2
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
2x2 + 3
Compute the derivative of f (x) = 2
x −4
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
2x2 + 3
Compute the derivative of f (x) = 2
x −4
f 0 (x)
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
2x2 + 3
Compute the derivative of f (x) = 2
x −4
0
f (x) =
Bernd Schröder
The Product and Quotient Rules
0
0
2x2 + 3 x2 − 4 − x2 − 4 2x2 + 3
2
(x2 − 4)
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
2x2 + 3
Compute the derivative of f (x) = 2
x −4
0
f (x) =
=
Bernd Schröder
The Product and Quotient Rules
0
0
2x2 + 3 x2 − 4 − x2 − 4 2x2 + 3
2
(x2 − 4)
4x x2 − 4 − 2x 2x2 + 3
2
(x2 − 4)
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
2x2 + 3
Compute the derivative of f (x) = 2
x −4
0
f (x) =
=
=
Bernd Schröder
The Product and Quotient Rules
0
0
2x2 + 3 x2 − 4 − x2 − 4 2x2 + 3
2
(x2 − 4)
4x x2 − 4 − 2x 2x2 + 3
2
(x2 − 4)
4x3 − 16x − 4x3 − 6x
2
(x2 − 4)
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
2x2 + 3
Compute the derivative of f (x) = 2
x −4
0
f (x) =
=
=
=
Bernd Schröder
The Product and Quotient Rules
0
0
2x2 + 3 x2 − 4 − x2 − 4 2x2 + 3
2
(x2 − 4)
4x x2 − 4 − 2x 2x2 + 3
2
(x2 − 4)
4x3 − 16x − 4x3 − 6x
2
(x2 − 4)
−22x
2
(x2 − 4)
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
2x2 + 3
Compute the derivative of f (x) = 2
x −4
0
f (x) =
=
=
=
0
0
2x2 + 3 x2 − 4 − x2 − 4 2x2 + 3
2
(x2 − 4)
4x x2 − 4 − 2x 2x2 + 3
2
(x2 − 4)
4x3 − 16x − 4x3 − 6x
2
(x2 − 4)
−22x
2
(x2 − 4)
(Leave the denominator as a power.)
Bernd Schröder
The Product and Quotient Rules
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Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) = 1 · ex
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) = 1 · ex + xex
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) = 1 · ex + xex
= ex (x + 1)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) = 1 · ex + xex
= ex (x + 1)
f 0 (1)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) = 1 · ex + xex
= ex (x + 1)
f 0 (1) = 2e
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) = 1 · ex + xex
= ex (x + 1)
f 0 (1) = 2e
f (1)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) =
=
f 0 (1) =
f (1) =
Bernd Schröder
The Product and Quotient Rules
1 · ex + xex
ex (x + 1)
2e
e
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) =
=
f 0 (1) =
f (1) =
y =
Bernd Schröder
The Product and Quotient Rules
1 · ex + xex
ex (x + 1)
2e
e
mx + b
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) =
=
f 0 (1) =
f (1) =
y =
e
Bernd Schröder
The Product and Quotient Rules
1 · ex + xex
ex (x + 1)
2e
e
mx + b
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) =
=
f 0 (1) =
f (1) =
y =
e =
Bernd Schröder
The Product and Quotient Rules
1 · ex + xex
ex (x + 1)
2e
e
mx + b
2e
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) =
=
f 0 (1) =
f (1) =
y =
e =
Bernd Schröder
The Product and Quotient Rules
1 · ex + xex
ex (x + 1)
2e
e
mx + b
2e · 1
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) =
=
f 0 (1) =
f (1) =
y =
e =
Bernd Schröder
The Product and Quotient Rules
1 · ex + xex
ex (x + 1)
2e
e
mx + b
2e · 1 + b
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) =
=
f 0 (1) =
f (1) =
y =
e =
b =
Bernd Schröder
The Product and Quotient Rules
1 · ex + xex
ex (x + 1)
2e
e
mx + b
2e · 1 + b
−e
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
f 0 (x) =
=
f 0 (1) =
f (1) =
y =
e =
b =
y =
Bernd Schröder
The Product and Quotient Rules
1 · ex + xex
ex (x + 1)
2e
e
mx + b
2e · 1 + b
−e
2ex − e
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
Find the Equation of the Tangent Line of
f (x) = xex at a = 1
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
f 0 (x)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
Bernd Schröder
The Product and Quotient Rules
1 · x2 + 4 − 2x · x
2
(x2 + 4)
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
=
Bernd Schröder
The Product and Quotient Rules
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
2
(x2 + 4)
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x)
=
=
2
2
2
(x + 4)
(x2 + 4)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
Bernd Schröder
The Product and Quotient Rules
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
-
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
-
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
-
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−2
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−2
Bernd Schröder
The Product and Quotient Rules
2
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−2
Bernd Schröder
The Product and Quotient Rules
2
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−2
Bernd Schröder
The Product and Quotient Rules
2
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−2
Bernd Schröder
The Product and Quotient Rules
2
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−3
−2
Bernd Schröder
The Product and Quotient Rules
2
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
f0
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
f0
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−−−
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
f0
−−−
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
+++
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
f0
−−−
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−−−
+++
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
f
f0
−−−
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−−−
+++
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
f decreasing
−−−
f0
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
−−−
+++
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
f decreasing
−−−
f0
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
increasing
+++
−−−
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
0
f (x) =
x1,2
f decreasing
−−−
f0
1 · x2 + 4 − 2x · x
2
(x2 + 4)
4 − x2
(2 − x)(2 + x) !
=
=
=0
2
2
2
(x + 4)
(x2 + 4)
= ±2
increasing
+++
decreasing
−−−
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
f
f0
decreasing
−−−
increasing
+++
decreasing
−−−
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
f
f0
decreasing
−−−
increasing
+++
decreasing
−−−
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
Find Where f (x) =
The Rules
Examples
x
is Increasing or
x2 + 4
Decreasing
f
f0
decreasing
−−−
increasing
+++
decreasing
−−−
−3
−2
Bernd Schröder
The Product and Quotient Rules
0
2
3
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
x 2 ex
Compute the Derivative of f (x) =
x+1
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
x 2 ex
Compute the Derivative of f (x) =
x+1
f 0 (x)
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
x 2 ex
Compute the Derivative of f (x) =
x+1
f 0 (x) =
Bernd Schröder
The Product and Quotient Rules
2xex + ex x2 (x + 1) − 1 · x2 ex
(x + 1)2
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
x 2 ex
Compute the Derivative of f (x) =
x+1
x + ex x2 (x + 1) − 1 · x2 ex
2xe
f 0 (x) =
(x + 1)2
ex 2x2 + x3 + 2x + x2 − x2 ex
=
(x + 1)2
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Derivation
The Rules
Examples
x 2 ex
Compute the Derivative of f (x) =
x+1
x + ex x2 (x + 1) − 1 · x2 ex
2xe
f 0 (x) =
(x + 1)2
ex 2x2 + x3 + 2x + x2 − x2 ex
=
(x + 1)2
ex x3 + 2x2 + 2x
=
(x + 1)2
Bernd Schröder
The Product and Quotient Rules
logo1
Louisiana Tech University, College of Engineering and Science