Question

given the function f(x) = 3x^2 sin 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, $u = 3x^{2}$ and $v=\sin x$.

Step2: Find $u'$

Differentiate $u = 3x^{2}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $u'=\frac{d}{dx}(3x^{2})=6x$.

Step3: Find $v'$

Differentiate $v=\sin x$ with respect to $x$. The derivative of $\sin x$ is $\cos x$, so $v'=\cos x$.

Step4: Calculate $f'(x)$

Substitute $u$, $u'$, $v$, and $v'$ into the product - rule formula: $f'(x)=u'v + uv'=6x\sin x+3x^{2}\cos x$.

Answer:

$6x\sin x + 3x^{2}\cos x$