Open MathNSWGitHub ↗

Differentiating trigonometric functions

Differentiate sin⁡x\sin x, cos⁡x\cos x and tan⁡x\tan x; apply the chain rule to differentiate sin⁡f(x)\sin f(x), cos⁡f(x)\cos f(x) and tan⁡f(x)\tan f(x).

Worked examples

Standard derivatives

Straightforward

Problem

Find dydx\dfrac{dy}{dx} for y=2sin⁡x−3cos⁡xy = 2\sin x - 3\cos x.

Chain rule with trigonometric functions

Moderate

Problem

Find ddx(cos⁡(4x−π))\dfrac{d}{dx}\left(\cos(4x - \pi)\right).

Finding stationary points

Challenging

Problem

Find the stationary points of y=sin⁡(2x)y = \sin(2x) for 0≤x≤π0 \leq x \leq \pi.

Practise

Q1·Straightforward
What is ddx(sin⁡x)\dfrac{d}{dx}(\sin x)?
Q2·Straightforward
What is ddx(cos⁡x)\dfrac{d}{dx}(\cos x)?
Q3·Straightforward
What is ddx(tan⁡x)\dfrac{d}{dx}(\tan x)?
Q4·Straightforward
Using the chain rule, ddx(sin⁡3x)=\dfrac{d}{dx}(\sin 3x) =
Q5·Straightforward
Find ddx(cos⁡5x)\dfrac{d}{dx}(\cos 5x).
Q6·Moderate
Find ddx(sin⁡(x2))\dfrac{d}{dx}\left(\sin(x^2)\right).
Q7·Moderate
Find the gradient of y=sin⁡xy = \sin x at x=π3x = \dfrac{\pi}{3}. Give an exact answer as a decimal to 2 decimal places.
Q8·Moderate
Find ddx(4cos⁡(x2))\dfrac{d}{dx}\left(4\cos\left(\dfrac{x}{2}\right)\right).
Q9·Moderate
Find ddx(tan⁡(2x+1))\dfrac{d}{dx}\left(\tan(2x + 1)\right).
Q10·Moderate
The function f(x)=3sin⁡(2x)f(x) = 3\sin(2x) has a stationary point in the interval 0<x<π0 < x < \pi. Find the xx-value (in radians) of the first stationary point to 3 decimal places.
Q11·Challenging
Find the gradient of y=cos⁡(x2)y = \cos(x^2) at x=πx = \sqrt{\pi}. Give your answer to 3 decimal places.
Q12·Challenging
Find d2ydx2\dfrac{d^2y}{dx^2} for y=sin⁡(3x)y = \sin(3x).
Q13·Challenging
A particle moves so that its displacement is x(t)=2cos⁡(3t)+1x(t) = 2\cos(3t) + 1 metres at time tt seconds. Find the velocity (in m/s) of the particle at t=π6t = \dfrac{\pi}{6} seconds.