Question

If $f\left(x\right)=2^{\sin x}$, then what is the derivative of $f\left(x\right)$?

• A. $2^{\sin x}\ln 2$
• B. $\left(\sin x\right)2^{\sin x-1}$
• C. $\left(\cos x\right)2^{\sin x-1}$
• D. None of the above

Answer 1

The correct option is D None of the above

$f\left(x\right)=2^{\sin x}$

$\Rightarrow \ln \left(f\left(x\right)\right)=\sin x\left(\ln 2\right)$

Differentiating with respect to $x,$ we have

$\frac{1}{f\left(x\right)}\frac{df\left(x\right)}{dx}=\ln 2\cos x\Rightarrow \frac{df\left(x\right)}{dx}=2^{\sin x}\ln 2\cos x$

None of the above is correct.