Combining the rules

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

Worked examples

Product rule

Straightforward

Problem

Find

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

Quotient rule

Moderate

Problem

Find

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

Chain rule inside a product

Challenging

Problem

Find

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

Practise

Open Math

Combining the rules

Calculus · MAV-12-04

Name:

Date:

Q1Straightforward

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. $\dfrac{u'v - uv'}{v^2}$

Q2Straightforward

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. $\dfrac{uv' - u'v}{v^2}$

Q3Moderate

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. $xe^x - e^x$

Q4Moderate

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. $x^2\cos x - 2x\sin x$

Q5Moderate

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. $\dfrac{xe^x - e^x}{x}$

Q6Moderate

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. $\dfrac{\cos x}{2x}$

Q7Moderate

Find the gradient of

y = x\ln x

x = e

. Give an exact numeric value.

Q8Moderate

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. $e^x(\cos x - \sin x)$

Q9Challenging

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. $\dfrac{1 + \ln x}{xe^x}$

Q10Challenging

Find the

x

-coordinate of the stationary point of

y = xe^{-x}

for

x > 0

. Give an exact answer.

Q11Challenging

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. $-e^x\cos(e^x)$

Q12Challenging

Find the second derivative

f''(x)

f(x) = x^2e^x

x = 0

Q13Challenging

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/combining-the-rules