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

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))