find the derivative of $f(x)=x^{7}cos x$ $f(x)=$

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u(x)v(x)$, then $y'=u'(x)v(x)+u(x)v'(x)$. Here, $u(x)=x^{7}$ and $v(x)=\cos x$.

Step2: Find $u'(x)$

Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, for $u(x)=x^{7}$, we have $u'(x)=7x^{6}$.

Step3: Find $v'(x)$

The derivative of $\cos x$ is $-\sin x$, so $v'(x)=-\sin x$.

Step4: Substitute into product - rule

$f'(x)=u'(x)v(x)+u(x)v'(x)=7x^{6}\cos x+x^{7}(-\sin x)$.

Answer:

$7x^{6}\cos x - x^{7}\sin x$