differentiate $f(x)=x^{8}cos x$. $f(x)=$

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = x^{8}$ and $v=\cos x$.

Step2: Differentiate $u$

The derivative of $u = x^{8}$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $u'=8x^{7}$.

Step3: Differentiate $v$

The derivative of $v=\cos x$ is $v'=-\sin x$.

Step4: Substitute $u$, $u'$, $v$, $v'$ into product - rule

$f'(x)=u'v + uv'=8x^{7}\cos x+x^{8}(-\sin x)$.

Answer:

$8x^{7}\cos x - x^{8}\sin x$