Question

if (f(x)=e^{x}sin x), then (f(x)=)
(a) (e^{x}cos x)
(b) (-e^{x}cos x)
(c) (e^{x}(sin x+cos x))
(d) (e^{x}(sin x - cos x))

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = e^{x}$ and $v=\sin x$.

Step2: Find derivatives of $u$ and $v$

The derivative of $u = e^{x}$ is $u'=e^{x}$, and the derivative of $v=\sin x$ is $v'=\cos x$.

Step3: Substitute into product - rule

$f'(x)=e^{x}\cdot\sin x+e^{x}\cdot\cos x=e^{x}(\sin x + \cos x)$

Answer:

C. $e^{x}(\sin x+\cos x)$