Lesson 40
Derivative of
ex
Examples
Derivative of
ln x
Examples
Section 4.3: Differentiation of Exponential and
Logarithmic Functions
April 23rd, 2014
Lesson 40
Derivative of
ex
Examples
Derivative of
ln x
Examples
In this lesson we will learn how to differentiate exponential and
logarithmic functions. We will incorporate all the dervative
rules that we have seen (power rule, quotient rule, chain rule)
in order to take the derivative of more complicated exponential
and logarithmic functions.
Lesson 40
The Derivative of e x
Derivative of
ex
Examples
Derivative of
ln x
Examples
For every real number x,
d
x
dx [e ]
= ex .
If u(x) is a differentiable function of x, then
d
u(x) ]
dx [e
=
du
dx
· e u(x) .
So e x is its own derivative. Notice that the second rule is really
just the chain rule where the outer function is e x and the inner
function is u(x). It says that to take the derivative of e u(x) , we
just multiply e u(x) by the derivative of u(x).
Lesson 40
Example
Derivative of
ex
Differentiate the given function.
Examples
Derivative of
ln x
Examples
f (x) = e 2x
This is really the composition of two functions where the
outer function is g (x) = e x and the inner function is
h(x) = 2x. Thus the derivative of f (x) is g 0 (h(x))h0 (x).
Since g 0 (x) = e x and h0 (x) = 2, we have
f 0 (x) = e 2x (2) = 2e 2x
This is also just the derivative rule given on the previous
page.
Lesson 40
f (x) = x 2 e x
Derivative of
ex
Examples
Derivative of
ln x
Here we must us the product rule, as f (x) is the product
of x 2 and e x .
f 0 (x) = (2x)e x + x 2 (e x ) = 2xe x + x 2 e x
Examples
f (x) = e x
2 +2x−1
f 0 (x) = (2x + 2)e x
2 +2x−1
f (x) = (x 2 + e 2x )3
Here we must use the chain rule twice, once for
(x 2 + e 2x )3 and once for e 2x .
f 0 (x) = 3(x 2 + e 2x )2 (2x + 2e 2x )
Lesson 40
The Derivative of ln x
Derivative of
ex
For all x > 0,
Examples
d
dx [ln x]
Derivative of
ln x
= x1 .
Examples
If u(x) is a differentiable function of x, then
d
dx [ln(u(x))]
=
1 du
u(x) dx
for u(x) > 0.
As before, the second derivative rule given here is just the
chain rule in the situation where the outer function is ln x.
Lesson 40
Example
Differentiate the given function.
Derivative of
ex
Examples
f (x) = ln(2x)
Derivative of
ln x
f 0 (x) =
Examples
1
2x (2)
=
2
2x
=
1
x
√
f (x) = x 3 ln(2 x)
√
f (x) = 3x ln(2 x) + x 3
0
1
√
2 x
√
1
= 3x 2 ln(2 x) + x 3
2x
2
√
x
= 3x 2 ln(2 x) +
2
2
1
√
x
Lesson 40
2
+1
f (x) = ln( xx−1
)
Derivative of
ex
Examples
Derivative of
ln x
Examples
(x − 1)2x − (x 2 + 1)
f (x) = 2 x +1
(x − 1)2
x−1
x − 1 x 2 − 2x − 1
= 2
x +1
(x − 1)2
2
x − 2x − 1
= 2
(x + 1)(x − 1)
1
0
Note that this is not the only way we can approach this
problem. If we want, we can first use the identity
2
+1
ln( xx−1
) = ln(x 2 + 1) − ln(x − 1)
Let’s continue with this on the next slide.
Lesson 40
Derivative of
ex
2
+1
) = ln(x 2 + 1) − ln(x − 1)
f (x) = ln( xx−1
0
f (x) =
Examples
1
2
x +1
2x −
1
x −1
Derivative of
ln x
Examples
=
2x
1
−
+1 x −1
x2
=
2x(x − 1) − (x 2 + 1)
(x 2 + 1)(x − 1)
=
x 2 − 2x − 1
(x 2 + 1)(x − 1)
Lesson 40
Derivative of
ex
Examples
Derivative of
ln x
Examples
h(t) =
e t +t
ln t
(ln t)(e t + 1) − (e t + t) 1t
h (t) =
(ln t)2
0
t
e t ln t + ln t − et − 1
=
(ln t)2
t
te ln t + t ln t − e t − t
=
t(ln t)2
Lesson 40
Derivative of
ex
Examples
g (u) = ln(u +
√
u 2 + 1)
1
√
g (u) =
u + u2 + 1
0
1 2
−1/2
1 + (u + 1)
(2u)
2
Derivative of
ln x
Examples
1
√
=
u + u2 + 1
1
√
=
u + u2 + 1
1
=√
2
u +1
u
√
1+
u2 + 1
√
u2 + 1 + u
√
u2 + 1
!