Question

If $f\left(x\right)=e^x\sin x$, then $f\text{'}\text{'}\left(x\right)$ is equal to

• A. $e^{6x}\sin 6x$
• B. $2e^x\cos x$
• C. $8e^x\sin x$
• D. $8e^x\cos x$

Answer 1

The correct option is B

$2e^x\cos x$

Explanation for the correct option.

Find the value of $f\text{'}\text{'}\left(x\right)$.

Differentiate both sides of the function $f\left(x\right)=e^x\sin x$ with respect to $x$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}\left(e^x\sin x\right)\\ & =& e^x\sin x+e^x\cos x\end{array}$

Now, differentiate the function $f\text{'}\left(x\right)$ with respect to $x$.

$\begin{array}{rcl}f\text{'}\text{'}\left(x\right)& =& \frac{d}{dx}\left(e^x\sin x+e^x\cos x\right)\\ & =& e^x\sin x+e^x\cos x+e^x\cos x+e^x\left(-\sin x\right)\\ & =& e^x\sin x+e^x\cos x+e^x\cos x-e^x\sin x\\ & =& 2e^x\cos x\end{array}$

Hence, the correct option is B.