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Worked solution

Collect implicit derivative terms with a constant-base exponential

The variables xx and yy are related by 3y=xy−sec⁡x3^y=xy-\sec x. Find dydx\dfrac{dy}{dx} in terms of xx and yy. marks

[2014/CJC/Promo/2]

CJC · Review Problems · Question 9

Answer

dydx=y−sec⁡xtan⁡x3yln⁡3−x\dfrac{dy}{dx}=\dfrac{y-\sec x\tan x}{3^y\ln3-x}.

Implicit differentiation

Differentiating gives 3yln⁡3 y′=y+xy′−sec⁡xtan⁡x3^y\ln3\,y\prime=y+xy\prime-\sec x\tan x. Thus (3yln⁡3−x)y′=y−sec⁡xtan⁡x(3^y\ln3-x)y\prime=y-\sec x\tan x, and dydx=y−sec⁡xtan⁡x3yln⁡3−x\dfrac{dy}{dx}=\dfrac{y-\sec x\tan x}{3^y\ln3-x} wherever the denominator is nonzero.

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Common mistakes

The derivative of xy has both a y term and an x dy/dx term.

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