Question

If $S=e^x\sin x$, find $\frac{d^{2}S}{d{x}^2}$

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

Answer 1

The correct option is C $2e^x\cos x$

Given, $S=e^x\sin x$
So, $\frac{d^{2}S}{d{x}^2}=\frac{d\left(\frac{dS}{dx}\right)}{dx}$
Thus, we have
$\frac{dS}{dx}=\frac{d\left(e^x\sin x\right)}{dx}=e^x\frac{d\sin x}{dx}+\sin x\frac{d{e}^x}{dx}$
$\Rightarrow e^x\cos x+e^x\sin x=e^x(\sin x+\cos x)$
Now, $\frac{d^{2}S}{d{x}^2}=\frac{d\left(e^x\right(\sin x+\cos x\left)\right)}{dx}$
$\Rightarrow e^x\frac{d(\sin x+\cos x)}{dx}+(\sin x+\cos x)\frac{d{e}^x}{dx}$
$\Rightarrow e^x(\cos x-\sin x)+e^x(\sin x+\cos x)$
$\Rightarrow e^x(\cos x-\sin x+\sin x+\cos x)=2e^x\cos x$