Sec 4 · Additional Mathematics

Differentiation of trigonometric, exponential and logarithmic functions

Practise differentiation of trigonometric, exponential and logarithmic functions with step-by-step help. Free account.

1. Trigonometric functions

Function	Derivative	Example
$\sin(ax + b)$	$a\cos(ax + b)$	$\dfrac{d}{dx}\left(\sin 3x\right) = 3\cos 3x$
$\cos(ax + b)$	$-a\sin(ax + b)$	$\dfrac{d}{dx}\left(\cos 2x\right) = -2\sin 2x$
$\tan(ax + b)$	$a\sec^{2}(ax + b)$	$\dfrac{d}{dx}\left(\tan 4x\right) = 4\sec^{2} 4x$

A power of sin: chain rule

\displaystyle y = \sin^{3} 2x

$\dfrac{dy}{dx} = 3\sin^{2} 2x \times 2\cos 2x$
$\dfrac{dy}{dx} = 6\sin^{2} 2x\cos 2x$

📖 Worked example

Given that $y = x^{2}\cos 3x$ , where $x$ is in radians, find $\dfrac{dy}{dx}$ .

Step 1 of 4

Which rule do you need?

Use the product rule.

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2. Exponential functions

Function	Derivative	Example
$e^{x}$	$e^{x}$
$e^{ax + b}$	$ae^{ax + b}$	$\dfrac{d}{dx}\left(e^{3x + 1}\right) = 3e^{3x + 1}$
$e^{f(x)}$	$f'(x)e^{f(x)}$	$\dfrac{d}{dx}\left(e^{x^{2}}\right) = 2xe^{x^{2}}$

Product rule, then the common factor

\displaystyle y = x^{2}e^{3x}

$\dfrac{dy}{dx} = x^{2} \cdot 3e^{3x} + e^{3x} \cdot 2x$
$\dfrac{dy}{dx} = xe^{3x}(3x + 2)$

📖 Worked example

Given that $y = x^{2}e^{3x}$ , find $\dfrac{dy}{dx}$ .

Step 1 of 4

Which rule do you need?

Use the product rule.

Practise this →

3. Logarithmic functions

Function	Derivative	Example
$\ln x$	$\dfrac{1}{x}$
$\ln(ax + b)$	$\dfrac{a}{ax + b}$	$\dfrac{d}{dx}\left(\ln(2x + 1)\right) = \dfrac{2}{2x + 1}$
$\ln f(x)$	$\dfrac{f'(x)}{f(x)}$	$\dfrac{d}{dx}\left(\ln(x^{2} + 1)\right) = \dfrac{2x}{x^{2} + 1}$

Use the laws of logarithms first

\displaystyle y = \ln \dfrac{x^{3}}{x + 1}

$y = 3\ln x - \ln(x + 1)$
$\dfrac{dy}{dx} = \dfrac{3}{x} - \dfrac{1}{x + 1}$

📖 Worked example

Given that $y = \ln\sqrt{2x + 1}$ , find $\dfrac{dy}{dx}$ .

Step 1 of 4

Write the square root as a power, then use a law of logarithms

y = \dfrac{1}{2}\ln(2x + 1)

Practise this →

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Practice questions by skill

1. Trigonometric functions

Function	Derivative	Example
$\sin(ax + b)$	$a\cos(ax + b)$	$\dfrac{d}{dx}\left(\sin 3x\right) = 3\cos 3x$
$\cos(ax + b)$	$-a\sin(ax + b)$	$\dfrac{d}{dx}\left(\cos 2x\right) = -2\sin 2x$
$\tan(ax + b)$	$a\sec^{2}(ax + b)$	$\dfrac{d}{dx}\left(\tan 4x\right) = 4\sec^{2} 4x$

A power of sin: chain rule

\displaystyle y = \sin^{3} 2x

$\dfrac{dy}{dx} = 3\sin^{2} 2x \times 2\cos 2x$
$\dfrac{dy}{dx} = 6\sin^{2} 2x\cos 2x$

📖 Worked example

Given that $y = x^{2}\cos 3x$ , where $x$ is in radians, find $\dfrac{dy}{dx}$ .

Step 1 of 4

Which rule do you need?

Use the product rule.

Practise this →

2. Exponential functions

Function	Derivative	Example
$e^{x}$	$e^{x}$
$e^{ax + b}$	$ae^{ax + b}$	$\dfrac{d}{dx}\left(e^{3x + 1}\right) = 3e^{3x + 1}$
$e^{f(x)}$	$f'(x)e^{f(x)}$	$\dfrac{d}{dx}\left(e^{x^{2}}\right) = 2xe^{x^{2}}$

Product rule, then the common factor

\displaystyle y = x^{2}e^{3x}

$\dfrac{dy}{dx} = x^{2} \cdot 3e^{3x} + e^{3x} \cdot 2x$
$\dfrac{dy}{dx} = xe^{3x}(3x + 2)$

📖 Worked example

Given that $y = x^{2}e^{3x}$ , find $\dfrac{dy}{dx}$ .

Step 1 of 4

Which rule do you need?

Use the product rule.

Practise this →

3. Logarithmic functions

Function	Derivative	Example
$\ln x$	$\dfrac{1}{x}$
$\ln(ax + b)$	$\dfrac{a}{ax + b}$	$\dfrac{d}{dx}\left(\ln(2x + 1)\right) = \dfrac{2}{2x + 1}$
$\ln f(x)$	$\dfrac{f'(x)}{f(x)}$	$\dfrac{d}{dx}\left(\ln(x^{2} + 1)\right) = \dfrac{2x}{x^{2} + 1}$

Use the laws of logarithms first

\displaystyle y = \ln \dfrac{x^{3}}{x + 1}

$y = 3\ln x - \ln(x + 1)$
$\dfrac{dy}{dx} = \dfrac{3}{x} - \dfrac{1}{x + 1}$

📖 Worked example

Given that $y = \ln\sqrt{2x + 1}$ , find $\dfrac{dy}{dx}$ .

Step 1 of 4

Write the square root as a power, then use a law of logarithms

y = \dfrac{1}{2}\ln(2x + 1)

Practise this →