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)!