Question #26703

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

Expert's answer

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.

(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.

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