To find the derivative of the given function $f(x) = 7x^2 \cos(x)$, we will use the product rule. The product rule states that if we have a function $f(x) = u(x)v(x)$, then its derivative is:

$f'(x) = u'(x)v(x) + u(x)v'(x)$

Here, $u(x) = 7x^2$ and $v(x) = \cos(x)$.

First, we find the derivatives of $u(x)$ and $v(x)$:

1. $u(x) = 7x^2$ $u'(x) = \frac{d}{dx}[7x^2] = 14x$

2. $v(x) = \cos(x)$ $v'(x) = \frac{d}{dx}[\cos(x)] = -\sin(x)$

Now we apply the product rule:

$f'(x) = u'(x)v(x) + u(x)v'(x)$

Substitute $u(x)$, $u'(x)$, $v(x)$, and $v'(x)$ into the product rule formula:

$f'(x) = (14x)(\cos(x)) + (7x^2)(-\sin(x))$

Simplify the expression:

$f'(x) = 14x \cos(x) - 7x^2 \sin(x)$

Thus, the derivative of $f(x) = 7x^2 \cos(x)$ is:

$f'(x) = 14x \cos(x) - 7x^2 \sin(x)$

Would you like more details or have any questions?

Here are 5 numbered questions for you to ask:

1. How do you apply the product rule in calculus?
2. Can you show an example of using the chain rule?
3. What is the derivative of trigonometric functions?
4. How do you simplify expressions involving trigonometric derivatives?
5. Can you explain how to find the second derivative of a function?

Tip: When using the product rule, carefully identify and differentiate each part of the product separately before combining the results.