Differential calculus

Extend differentiation to exponential, logarithmic and trigonometric functions; apply the product, quotient and chain rules; find second derivatives.

Calculus · MAV-12-04

Sub-topics

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.

Differentiating trigonometric functions

Differentiate

\sin x

\cos x

and

\tan x

; apply the chain rule to differentiate

\sin f(x)

\cos f(x)

and

\tan f(x)

Combining the rules

Apply the product rule, quotient rule and chain rule to differentiate combinations of exponential, logarithmic and trigonometric functions; find second derivatives.

Open Math

Differential calculus

Calculus · MAV-12-04

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Differentiating exponential and logarithmic functions

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?

Differentiating trigonometric functions

Q14Straightforward

What is

\dfrac{d}{dx}(\sin x)

A. $-\sin x$
B. $\cos x$
C. $-\cos x$
D. $\sec^2 x$

Q15Straightforward

What is

\dfrac{d}{dx}(\cos x)

A. $\sin x$
B. $\sec^2 x$
C. $-\cos x$
D. $-\sin x$

Q16Straightforward

What is

\dfrac{d}{dx}(\tan x)

A. $\sec x$
B. $\cot x$
C. $\sec^2 x$
D. $-\sec^2 x$

Q17Straightforward

Using the chain rule,

\dfrac{d}{dx}(\sin 3x) =

A. $\cos 3x$
B. $3\cos 3x$
C. $-3\cos 3x$
D. $\dfrac{\cos 3x}{3}$

Q18Straightforward

Find

\dfrac{d}{dx}(\cos 5x)

A. $5\sin 5x$
B. $-\sin 5x$
C. $-5\sin 5x$
D. $5\cos 5x$

Q19Moderate

Find

\dfrac{d}{dx}\left(\sin(x^2)\right)

A. $\cos(x^2)$
B. $2x\cos(x^2)$
C. $-2x\cos(x^2)$
D. $2\cos(2x)$

Q20Moderate

Find the gradient of

y = \sin x

x = \dfrac{\pi}{3}

. Give an exact answer as a decimal to 2 decimal places.

Q21Moderate

Find

\dfrac{d}{dx}\left(4\cos\left(\dfrac{x}{2}\right)\right)

A. $-2\sin\left(\dfrac{x}{2}\right)$
B. $2\sin\left(\dfrac{x}{2}\right)$
C. $-4\sin\left(\dfrac{x}{2}\right)$
D. $4\sin\left(\dfrac{x}{2}\right)$

Q22Moderate

Find

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

A. $\sec^2(2x+1)$
B. $2\sec^2(2x+1)$
C. $2\tan(2x+1)$
D. $-2\sec^2(2x+1)$

Q23Moderate

The function

f(x) = 3\sin(2x)

has a stationary point in the interval

0 < x < \pi

. Find the

x

-value (in radians) of the first stationary point to 3 decimal places.

Q24Challenging

Find the gradient of

y = \cos(x^2)

x = \sqrt{\pi}

. Give your answer to 3 decimal places.

Q25Challenging

Find

\dfrac{d^2y}{dx^2}

for

y = \sin(3x)

A. $3\cos(3x)$
B. $-3\cos(3x)$
C. $-9\sin(3x)$
D. $9\sin(3x)$

Q26Challenging

A particle moves so that its displacement is

x(t) = 2\cos(3t) + 1

metres at time

t

seconds. Find the velocity (in m/s) of the particle at

t = \dfrac{\pi}{6}

seconds.

Combining the rules

Q27Straightforward

State the product rule. If

y = u(x)\,v(x)

, then

\dfrac{dy}{dx} =

A. $u'v'$
B. $u'v + uv'$
C. $u'v - uv'$
D. $\dfrac{u'v - uv'}{v^2}$

Q28Straightforward

State the quotient rule. If

y = \dfrac{u(x)}{v(x)}

, then

\dfrac{dy}{dx} =

A. $\dfrac{u'v + uv'}{v^2}$
B. $\dfrac{u'v - uv'}{v^2}$
C. $\dfrac{u'}{v'}$
D. $\dfrac{uv' - u'v}{v^2}$

Q29Moderate

Use the product rule to find

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

A. $e^x$
B. $xe^x$
C. $e^x + xe^x$
D. $xe^x - e^x$

Q30Moderate

Use the product rule to find

\dfrac{d}{dx}\left(x^2\sin x\right)

A. $2x\cos x$
B. $x^2\cos x + 2x\sin x$
C. $2x\sin x + x^2\cos x$
D. $x^2\cos x - 2x\sin x$

Q31Moderate

Use the quotient rule to find

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

for

x \neq 0

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

Q32Moderate

Find

\dfrac{d}{dx}\left(\dfrac{\sin x}{x^2}\right)

A. $\dfrac{x^2\cos x - 2x\sin x}{x^4}$
B. $\dfrac{x\cos x - 2\sin x}{x^3}$
C. $\dfrac{x^2\cos x + 2x\sin x}{x^4}$
D. $\dfrac{\cos x}{2x}$

Q33Moderate

Find the gradient of

y = x\ln x

x = e

. Give an exact numeric value.

Q34Moderate

Find

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

A. $e^x\cos x - e^x\sin x$
B. $e^x\sin x - e^x\cos x$
C. $e^x(\cos x + \sin x)$
D. $e^x(\cos x - \sin x)$

Q35Challenging

Differentiate

y = \dfrac{\ln x}{e^x}

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

Q36Challenging

Find the

x

-coordinate of the stationary point of

y = xe^{-x}

for

x > 0

. Give an exact answer.

Q37Challenging

Find

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

A. $\cos(e^x)$
B. $e^x\cos(e^x)$
C. $e^x\sin(e^x)$
D. $-e^x\cos(e^x)$

Q38Challenging

Find the second derivative

f''(x)

f(x) = x^2e^x

x = 0

Q39Challenging

A function is defined as

y = \dfrac{\cos x}{1 + \sin x}

. Which expression equals

\dfrac{dy}{dx}

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

Worked solutions and answers at openmath.au/year-12/advanced/differential-calculus