What is the coefficient of 30 sin(3x) cos(3x)(4 + sin^2(3x))^4 when finding the derivative of y = (4 + sin^2(3x))^5?

Steps to Solve

1. Identify the function and rules to use

First, identify the function you need to differentiate and the parts of it that will require the chain rule and product rule. For example, if the function is $f(x) = g(h(x)) \cdot k(x)$, you need to differentiate both $g(h(x))$ and $k(x)$.

2. Differentiate using the product rule

Recall the product rule formula: if $u(x)$ and $v(x)$ are two functions, then the derivative is given by:

$$ (uv)' = u'v + uv' $$

In our case, let $u = g(h(x))$ and $v = k(x)$. Differentiate $u$ and $v$.

3. Apply the chain rule to the first function

To differentiate $u = g(h(x))$, apply the chain rule:

$$ u' = g'(h(x)) \cdot h'(x) $$

Make sure to calculate $g'(h(x))$ and $h'(x)$ separately.

4. Compute the derivative

Now that you have $u'$ and $v'$, plug these into the product rule formula:

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

5. Combine and simplify

Finally, combine the results from the previous step. Simplify the expression to find the coefficient you are interested in.