Section 3.1 Derivatives of Power functions and Exponential functions
Reminders:
As seen before the derivative of f ( x )  x 2 is f ' ( x )  2 x .
The derivative of a line is the slope of the line.
The linear rule says (af+bg)' (x)= af '(x) + bg'(x). That allows us to take the derivative of
any quadratic.
f (x)  3x
Example:
2
 15 x  12
f ' ( x )   3 ( 2 x )  15   6 x  15
Power rule: For
d
n  0,
n
(x
n 1
)  nx
n
f (x)  x ,
ie for
n 1
f ' ( x )  nx
dx
d
3
(x )  3x
d
2
dx
Examples:
(x
10
)  10 x
9
dx
1
d
2
(x

1
) 
dx
1
d
2
x
2
(
dx
1
d)
) 
x
(x
1
2
)  x
1

dx
2
x
The linear rule applied to polynomials:
 7x
2
f '( x )  2(4 x )  8(3 x
2
f (x)  2x
4
3
 8x
3
 3x  2
3
)  7(2 x )  3  8 x
 24 x
2
 14 x  3
Examples:
d
5
( 12 x
 4x
3
4
 13 x  8 )  12 ( 5 x
)  4(3 x
2
)  13  60 x
4
 12 x
2
 13
dx
Examples: Fractional and negative exponents:
1
f (x) 
3
x  x
1
f '(x) 
3

2
3
x
d
and
3
1
(
dx
2
x
d
) 
(x
2
)  2 x
dx
(4
3
x 
dx
5
x
2

4
) 
2
3
x

10
3
x
3
When the function is a quotient and the denominator is a power of x:
d
Example:
(
x
2
 7x  9
dx
) 
x
 1 9x
2
 1
d
dx
(
x
2
1
x
) 
d
(
x
2
dx
x
3 2
 x

7x
9

x
) 
x
d
(x  7  9x
dx
9
x
Example:
d
2
(x
1 2
dx
Derivatives of Exponential functions:
) 
3
2
x
1 2

1
2
x
3 2

 2
x
Applying the linear rule:
d
3
1
)
3
d
(a
x
)  a
x
d
ln a
dx
(e
x
)  e
x
dx
If the function f(x) is equal to its derivative, then f(x) =
No other functions have this property.
More examples:
f (x)  4x
f (x) 
3
 9(2
x  5e
x
f ' ( x )  12 x
)
2
 9(2
x
Ae
x
for some constant, A.
) ln 2
x
Find the equation of the tangent line to the graph of
f (x)  4x
2

2
e
x
at
x 1.
e
f (1 )  6
f '( x)  8 x 
2
e
x
f ' ( 1 )  10
e
y  10 ( x  1 )  6
One rule, I call the shift rule, you will not find in the book until we get to the chain rule.
But it saves work until we do the chain rule.
Suppose you shift f(x) to the right 2 units. You don't change the shape of the graph. The
tangent line should shift right 2 units also, but have the same slope.
The shift rule says
d
( f ( x  a ))  f ' ( x  a )
dx
Examples:
f (x)  (x  2)
g(x) 
x 1
3
f ' ( x )  3( x  2 )
g '( x) 
1

( x  1)
2
1
2
2
This only works for a shift. It does not work for any other composition.