Question

given the function f(x)=5x^3 cos x, find f(x) in any form.

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = 5x^{3}$ and $v=\cos x$.

Step2: Differentiate $u$

Differentiate $u = 5x^{3}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we get $u'=\frac{d}{dx}(5x^{3})=15x^{2}$.

Step3: Differentiate $v$

Differentiate $v=\cos x$ with respect to $x$. We know that $\frac{d}{dx}(\cos x)=-\sin x$.

Step4: Substitute into product - rule

Substitute $u = 5x^{3}$, $u' = 15x^{2}$, $v=\cos x$, and $v'=-\sin x$ into the product - rule $y'=u'v + uv'$.
$f'(x)=15x^{2}\cos x+5x^{3}(-\sin x)$.
$f'(x)=15x^{2}\cos x - 5x^{3}\sin x$.

Answer:

$f'(x)=15x^{2}\cos x - 5x^{3}\sin x$