Question

find the derivative of $x^{3}cos x$

Explanation:

Step1: Apply product - rule

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

Step2: Find $u^\prime$

Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, for $u = x^{3}$, we have $u^\prime=\frac{d}{dx}(x^{3}) = 3x^{2}$.

Step3: Find $v^\prime$

The derivative of $\cos x$ is $v^\prime=\frac{d}{dx}(\cos x)=-\sin x$.

Step4: Substitute into product - rule

$y^\prime=(x^{3})^\prime\cos x+x^{3}(\cos x)^\prime=3x^{2}\cos x - x^{3}\sin x$.

Answer:

$3x^{2}\cos x - x^{3}\sin x$