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: Recall product - rule

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

Step2: Find derivatives of $u(x)$ and $v(x)$

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

Step3: Apply the product - rule

$f^\prime(x)=u^\prime(x)v(x)+u(x)v^\prime(x)=e^{x}\sin x+e^{x}\cos x=e^{x}(\sin x + \cos x)$.

Answer:

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