Porisma

Worked solution

Three chain-rule derivatives

Differentiate each of the following with respect to x.

(i) esec⁡xe^{\sec x} mark

(ii) tan⁡−1(x2)\tan^{-1}(x^2) marks

(iii) cos⁡4(2x)\cos^4(2x) marks

[2014/PJC/Promo/1]

CJC · Review Problems · Question 1

Answer

(i) (sec⁡xtan⁡x)esec⁡x(\sec x\tan x)e^{\sec x}; (ii) 2x/(1+x4)2x/(1+x^4); (iii) −8cos⁡3(2x)sin⁡(2x)-8\cos^3(2x)\sin(2x).

(i)

ddxesec⁡x=(sec⁡xtan⁡x)esec⁡x\dfrac{d}{dx}e^{\sec x}=(\sec x\tan x)e^{\sec x}. The derivative of the inner secant supplies the factor sec x tan x.

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(ii)

ddxtan⁡−1(x2)=2x1+(x2)2=2x1+x4\dfrac{d}{dx}\tan^{-1}(x^2)=\dfrac{2x}{1+(x^2)^2}=\dfrac{2x}{1+x^4}.

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(iii)

ddxcos⁡4(2x)=4cos⁡3(2x)[−sin⁡(2x)] 2=−8cos⁡3(2x)sin⁡(2x)\dfrac{d}{dx}\cos^4(2x)=4\cos^3(2x)[-\sin(2x)]\,2=-8\cos^3(2x)\sin(2x).

How did this part go?

Common mistakes

Here tan⁻¹ is arctangent, not the reciprocal of tan. Work in radians.

Read the hint