find the derivative of $f(x)=e^{x}cos x$.

Question

Accepted Answer

# Explanation:
## Step1: Apply the product rule
The product rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = e^{x}$ and $v=\cos x$.
## Step2: Find the derivatives of $u$ and $v$
The derivative of $u = e^{x}$ is $u^\prime=e^{x}$. The derivative of $v=\cos x$ is $v^\prime=-\sin x$.
## Step3: Substitute into the product rule formula
$$
\begin{align*}
f^\prime(x)&=(e^{x})^\prime\cos x+e^{x}(\cos x)^\prime\
&=e^{x}\cos x+e^{x}(-\sin x)\
&=e^{x}(\cos x - \sin x)
\end{align*}
$$

# Answer:
$f^\prime(x)=e^{x}(\cos x - \sin x)$

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QUESTION IMAGE

Question

Explanation:

Step1: Apply the product rule

Step2: Find the derivatives of \(u\) and \(v\)

Step3: Substitute into the product rule formula

Answer: