Q: If f(x) = (-3x⁵+2x⁴+5x)²⁰(ln(2x²+2))²⁰, find f'(x).

A: The main rule here is the product rule. f(x) is equal to two functions multiplied together, I will call them g(x) and h(x).

So, f(x) = g(x)*h(x)

The product rules states that if f(x) = g(x)*h(x) then f'(x) = g'(x)*h(x) + g(x)*h'(x).

Now, all I must do is establish g(x), h(x), g'(x), h'(x) and plug each piece into the formula above and I am done!

From the problem we have:

g(x) = (-3x⁵+2x⁴+5x)²⁰

h(x) = (ln(2x²+2))²

Now, take the derivative of each piece:

g'(x) = 20(-3x⁵+2x⁴+5x)¹⁹ * (-15x⁴+8x³+5) — This was found using the chain rule: the derivative of the outside times the derivative of the inside!

h'(x) = 2(ln(2x²+2)) * 1/(2x²+2) * (4x) — A three part chain rule!

Now, plug in g, h, g’ and h’ into the product rule formula and you have the derivative of f'(x)!