Question

Use the chain rule to find the derivative of the function: y=(5x^3+4)^2. y'=?

Answer 1

The chain rule means that if f and g are twodifferentiable functions, then the derivative of their composition

(f(g(x))' = f'(g(x)) * g'(x).
In our case, denote
f(x) = x^2,
g(x) = 5x^3+4Then
y(x) = f(g(x)).

Notice that
f'(x) = (x^2)' =2x
g'(x) = (5x^3+4)'= 15 x^2 whence
y'(x) = f'(g(x)) * g'(x) = 2(5x^3+4) * 15 x^2

Simplifying this expression we get
y'(x) = 2(5x^3+4) * 15 x^2
= (10x^3 + 8)* 15 x^2
= 150x^5 +120x^2For additional practice you can watch our videos on chain rule, you'll find theoretical explanations and examples showing all the steps.