Question

find the derivative of the following function.
y = e^{-x} sin x
dy/dx = □

Explanation:

Step1: Apply product - rule

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

Step2: Find $u^\prime$

Differentiate $u = e^{-x}$ using the chain - rule. If $y = e^{f(x)}$, then $y^\prime=f^\prime(x)e^{f(x)}$. For $u = e^{-x}$, $f(x)=-x$ and $f^\prime(x)=-1$, so $u^\prime=-e^{-x}$.

Step3: Find $v^\prime$

Differentiate $v = \sin x$. The derivative of $\sin x$ with respect to $x$ is $\cos x$, so $v^\prime=\cos x$.

Step4: Substitute into product - rule

$y^\prime=u^\prime v+uv^\prime=-e^{-x}\sin x + e^{-x}\cos x=e^{-x}(\cos x-\sin x)$

Answer:

$e^{-x}(\cos x - \sin x)$