Porisma

Worked solution

Differentiate an exponential of a trigonometric function implicitly

Given that y3+etan⁡y=cos⁡(2x+π3)y^3+e^{\tan y}=\cos\left(2x+\dfrac\pi3\right), find dydx\dfrac{dy}{dx} in terms of xx and yy. marks

[2016/CJC/Promo/1]

CJC · Review Problems · Question 8

Answer

dydx=−2sin⁡(2x+π/3)3y2+sec⁡2y etan⁡y\dfrac{dy}{dx}=-\dfrac{2\sin(2x+\pi/3)}{3y^2+\sec^2y\,e^{\tan y}}.

Implicit differentiation

Differentiate both sides with respect to xx: 3y2y′+sec⁡2y etan⁡yy′=−2sin⁡(2x+π/3)3y^2y\prime+\sec^2y\,e^{\tan y}y\prime=-2\sin(2x+\pi/3). Collect the two terms containing y′y\prime to obtain dydx=−2sin⁡(2x+π/3)3y2+sec⁡2y etan⁡y\dfrac{dy}{dx}=-\dfrac{2\sin(2x+\pi/3)}{3y^2+\sec^2y\,e^{\tan y}}.

How did this part go?

Common mistakes

Each differentiation through y introduces dy/dx; the exponential contributes another inner derivative.

Read the hint