I don't think I fully understand the question. If I knew f(x) I would use the chain rule or product rule, but since I don't, how do I know what to do? Thanks for any help I appreciate it.
$\endgroup$ 11 Answer
$\begingroup$You can use the product rule for this provided $f$ is differentiable. Write your function as
$$9 \cdot x^2 \cdot f(x)$$
Then by linearity,
$$\frac{d}{dx} 9x^2f(x) = 9 \cdot \frac{d}{dx} \left( x^2 \cdot f(x) \right)$$
You can use the product rule here as normal.
Example: Say we want to find the derivative of $\sin(x)g(x)$ where $g$ is a differentiable function. By the product rule,
$$\frac{d}{dx} \sin(x)g(x) = \cos(x)g(x) + \sin(x)g'(x)$$
More generally, as long as $f,g$ are differentiable,
$$\frac{d}{dx} f(x)g(x) = f'(x)g(x) + f(x)g'(x)$$
In our example, we simply set $f(x) = \sin(x)$. In your case, you could set $g(x) = x^2$, as another means of looking at it.
$\endgroup$
