Question

Differentiate 9^tan⁡3x with respect to x.

(a) 9^tan⁡3x (3 log⁡9 sec^2⁡x)

(b) 9^tan⁡3x (3 log⁡3 sec^2⁡⁡x)

(c) 9^tan⁡3x (3 log⁡9 sec⁡x)

(d) -9^tan⁡3x (3 log⁡9 sec^2⁡⁡x)

The question was asked in semester exam.

Origin of the question is Logarithmic Differentiation topic in portion Continuity and Differentiability of Mathematics – Class 12

Answer 1

The correct option is (a) 9^tan⁡3x (3 log⁡9 sec^2⁡x)

For explanation: Consider y=9^tan⁡3x

Applying log on both sides, we get

log⁡y=log⁡9^tan⁡3x

Differentiating both sides with respect to x, we get

\(\frac{1}{y} \frac{dy}{dx}=\frac{d}{dx} \)(tan⁡3x.log⁡9)

\(\frac{1}{y} \frac{dy}{dx}=\frac{d}{dx} \,(tan⁡3x) \,log⁡9+\frac{d}{dx} \,(log⁡9).tan3x \,(∵ Using \,u.v=u’ \,v+uv’)\)

\(\frac{dy}{dx}\)=y(3 sec^2⁡⁡x.log⁡9+0)

\(\frac{dy}{dx}\)=9^tan⁡3x (3 log⁡9 sec^2⁡x)