Differentiating exponential and logarithmic functions

Differentiate

e^x

e^{f(x)}

\ln x

and

\ln f(x)

using the standard results and the chain rule.

Worked examples

Differentiating $e^{f(x)}$ using the chain rule

Straightforward

Problem

Find

\dfrac{d}{dx}\left(e^{4x - 1}\right)

Differentiating $\ln f(x)$ using the chain rule

Moderate

Problem

Find

\dfrac{d}{dx}\left(\ln(x^2 + 3x)\right)

Finding a gradient and tangent equation

Challenging

Problem

Find the equation of the tangent to

y = 3e^{2x}

at the point where

x = 0

Practise

Open Math

Differentiating exponential and logarithmic functions

Calculus · MAV-12-04

Name:

Date:

Q1Straightforward

What is

\dfrac{d}{dx}\left(e^x\right)

A. $xe^{x-1}$
B. $e^x$
C. $e^{x+1}$
D. $\ln x$

Q2Straightforward

What is

\dfrac{d}{dx}\left(\ln x\right)

A. $e^x$
B. $\dfrac{1}{x^2}$
C. $\dfrac{1}{x}$
D. $\ln x + 1$

Q3Straightforward

Using the chain rule,

\dfrac{d}{dx}\left(e^{3x}\right) =

A. $e^{3x}$
B. $3e^{3x}$
C. $\dfrac{1}{3}e^{3x}$
D. $3xe^{3x-1}$

Q4Straightforward

Find

\dfrac{d}{dx}\left(\ln(5x)\right)

A. $\dfrac{5}{x}$
B. $\dfrac{1}{5x}$
C. $\dfrac{1}{x}$
D. $5\ln x$

Q5Moderate

Find

\dfrac{d}{dx}\left(e^{x^2 + 1}\right)

A. $e^{x^2+1}$
B. $2x\,e^{x^2}$
C. $2x\,e^{x^2+1}$
D. $(x^2+1)\,e^{x^2}$

Q6Moderate

Find

\dfrac{d}{dx}\left(\ln(x^2 + 4)\right)

A. $\dfrac{2x}{x^2+4}$
B. $\dfrac{1}{x^2+4}$
C. $\dfrac{2}{x^2+4}$
D. $\dfrac{x}{x^2+4}$

Q7Moderate

Find the gradient of

y = e^{2x}

x = 0

. Give an exact answer.

Q8Moderate

The function

f(x) = \ln(3x - 6)

is defined for

x > 2

. Find

f'(x)

evaluated at

x = 3

Q9Moderate

Which of the following is

\dfrac{d}{dx}\left(5e^{-2x}\right)

A. $-10e^{-2x}$
B. $10e^{-2x}$
C. $5e^{-2x}$
D. $-5e^{-2x}$

Q10Moderate

Find the

x

-coordinate of the stationary point of

y = e^x - 3x

. Give an exact answer.

Q11Challenging

Find

\dfrac{d}{dx}\left(\ln\left(\dfrac{x+1}{x-1}\right)\right)

for

x > 1

. Simplify fully.

A. $\dfrac{-2}{x^2 - 1}$
B. $\dfrac{2}{x^2 - 1}$
C. $\dfrac{1}{x+1} - \dfrac{1}{x-1}$
D. $\dfrac{2}{(x+1)(x-1)}$

Q12Challenging

The tangent to

y = \ln x

at the point where

x = e

has equation

y = mx + c

. Find the

y

-intercept

c

Q13Challenging

A population of bacteria grows according to

P(t) = 500e^{0.3t}

, where

t

is time in hours. At what time (in hours, to 2 decimal places) is the rate of growth equal to 600 bacteria per hour?

Worked solutions and answers at openmath.au/year-12/advanced/differential-calculus/differentiating-exponential-and-logarithmic-functions

Differentiating exponential and logarithmic functions

Worked examples

Differentiating ef(x)e^{f(x)}ef(x) using the chain rule

Differentiating ln⁡f(x)\ln f(x)lnf(x) using the chain rule

Finding a gradient and tangent equation

Practise

Differentiating exponential and logarithmic functions

Differentiating $e^{f(x)}$ using the chain rule

Differentiating $\ln f(x)$ using the chain rule