Derivatives
What is a derivative?
• Mathematically, it is the
slope of the tangent line at
dy y rise
a given pt.


dx x run
• Scientifically, it is the
instantaneous velocity of a
particle along a line at
time, t.
• Or the instantaneous rate
of change of a fnc. at a pt.
Formal Definition of a Derivative:
f  a  h   f  a  is called the derivative of f at a.
lim
h 0
h
We write:
f   x   lim
h 0
f a  h  f a
h
“The derivative of f with respect to x is …”
There are many ways to write the derivative of y
 f  x

f  x
“f prime x”
y
“y prime”
or
“the derivative of f with respect
to x”
dy
dx
“dee why dee ecks”
or
“the derivative of y with
respect to x”
df
dx
“dee eff dee ecks”
or
“the derivative of f with
respect to x”
d
f  x  “dee dee ecks uv eff uv ecks” or “the derivative
dx
of f of x”
( d dx of f of x )

dy does not mean d times y !
dx does not mean d times x !

dy
does not mean dy  dx !
dx
(except when it is convenient to think of it as division.)
df
does not mean df  dx !
dx
(except when it is convenient to think of it as division.)

d
d
f  x  does not mean
times f  x  !
dx
dx
(except when it is convenient to treat it that way.)

A function is differentiable if it has a
derivative everywhere in its domain. The
limit must exist and it must be continuous
and smooth. Functions on closed
intervals must have one-sided
derivatives defined at the end points.
p
4
3
2
y  f  x
1
0
3
The derivative
is the slope of
the original
function.
1
2
3
4
5
6
7
8
9
2
The derivative is defined at the end points
of a function on a closed interval.
1
0
-1
-2
1
2
3
4
5
6
7
8
9
y  f  x

6
5
y  x 3
2
4
3
2
1
-3
-2
-1 0
-1
1
x
2
3
y  lim
 x  h
2
h 0
-2

3 x 3
2

h
-3
6
5
4
3
2
1
-3 -2 -1 0
-1
-2
-3
-4
-5
-6
x  2 xh  h  x
y  lim
h 0
h
2
1 2 3
x
2
y  lim 2 x  h
2
0
h 0
y  2 x

1.
2.
3.
4.
5.
6.
7.
8.
DIFFERENTIATION RULES:
If f(x) = c, where c is a constant, then f’(x) = 0
If f(x) = c*g(x), then f’(x) = c*g’(x)
If f(x) = xn, then f’(x) = nxn-1
SUM RULE:
If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x)
DIFFERENCE RULE:
If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x)
PRODUCT RULE:
If f(x) = g(x) * h(x), then f’(x) = g’(x)*h(x) +
h’(x)*g(x)
QUOTIENT RULE:
g ' ( x ) * h( x )  h' ( x ) * g ( x )
f(x) = g ( x ) , then f’(x) =
(h( x)) 2
h( x )
CHAIN RULE:
If f(x) = g(h(x)), then f’(x) = g’(x)*h’(x)
Derivatives to memorize:
•
•
•
•
•
•
•
•
•
If f(x) = sin x, then f’(x) = cos x
If f(x) = cos x, then f’(x) = -sin x
If f(x) = tan x, then f’(x) = sec2x
If f(x) = cot x, then f’(x) = -csc2x
If f(x) = sec x, then f’(x) = secxtanx
If f(x) = csc x, then f’(x) = -cscxcotx
If f(x) = ex, then f’(x) = ex
If f(x) = ln x, then f’(x) = 1/x
If f(x) = ax, then f’(x) = (ln a) * ax
Examples: Find f’(x) if
1.
2.
3.
4.
5.
f(x) = 5
f(x) = x2 – 5
f(x) = 6x3+5x2+9x+3
f(x) = (3x+4)(2x2-3x+5)
f(x) = 4
x2
6. f(x) =
3x  5
2x  3
7. f(x) = (3x2+5x-2)8
8. f(x)= 5x 2  3x 1