Question

use the product rule to differentiate the function. f(x) = x^5 cos(x)

Explanation:

Step1: Recall product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = x^{5}$ and $v=\cos(x)$.

Step2: Differentiate $u$ and $v$

The derivative of $u = x^{5}$ with respect to $x$ is $u^\prime=\frac{d}{dx}(x^{5}) = 5x^{4}$ (using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$). The derivative of $v=\cos(x)$ with respect to $x$ is $v^\prime=\frac{d}{dx}(\cos(x))=-\sin(x)$.

Step3: Apply product - rule

Substitute $u$, $u^\prime$, $v$, and $v^\prime$ into the product - rule formula. $f^\prime(x)=u^\prime v+uv^\prime=5x^{4}\cos(x)+x^{5}(-\sin(x))$.

Answer:

$f^\prime(x)=5x^{4}\cos(x)-x^{5}\sin(x)$