Question

Differentiate (3 cos⁡x)^x with respect to x.

(a) (3 cos⁡x)^x (log⁡(3 cos⁡x)+x tan⁡x)

(b) (3 cos⁡x)^x (log⁡(3 cos⁡x)+tan⁡x)

(c) (cos⁡x)^x (log⁡(3 cos⁡x)-x tan⁡x)

(d) (3 cos⁡x)^x (log⁡(3 cos⁡x)-x tan⁡x)

This question was posed to me in semester exam.

Enquiry is from Logarithmic Differentiation in chapter Continuity and Differentiability of Mathematics – Class 12

Answer 1

Correct option is (d) (3 cos⁡x)^x (log⁡(3 cos⁡x)-x tan⁡x)

The best I can explain: Consider y=(3 cos⁡x)^x

Applying log on both sides, we get

log⁡y=log⁡(3 cos⁡x)^x

log⁡y=x log⁡(3 cos⁡x)

log⁡y=x(log⁡3+log⁡(cos⁡x))

Differentiating both sides with respect to x, we get

\(\frac{1}{y} \frac{dy}{dx} =\frac{d}{dx} (x log⁡3)+\frac{d}{dx} (x) log⁡(cos⁡x)+\frac{d}{dx}(log⁡(cos⁡x)).x\)

\(\frac{1}{y} \frac{dy}{dx}=log⁡3+log⁡(cos⁡x)+\frac{1}{cos⁡x}.-sin⁡x.x\)

\(\frac{1}{y} \frac{dy}{dx}\)=log⁡3+log⁡(cos⁡x)-x tan⁡x

\(\frac{dy}{dx}\)=y(log⁡(3 cos⁡x)-x tan⁡x)

\(\frac{dy}{dx}\)=(3 cos⁡x)^x (log⁡(3 cos⁡x)-x tan⁡x)