How do you use the chain rule to differentiate y=(x^2+5x)^2+2(x^3-5x)^3?

1 Answer
Oct 9, 2016

(dy)/(dx)=2(2x+5)(x^2+5x)+6(3x^2-5)(x^3-5x)^2

Explanation:

Chain rule: (dy)/(dx)=(dy)/(du)*(du)/(dx)

We do this twice to derive both (x^2+5x)^2 and 2(x^3-5x)^3

d/(dx)(x^2+5x)^2: Let u=x^2+5x, then (du)/(dx)=2x+5
(dy)/(du)=2(x^2+5x)
So (dy)/(dx)=2(2x+5)(x^2+5x)

d/(dx)2(x^3-5x)^3: Let u=x^3-5x, then (du)/(dx)=3x^2-5
(dy)/(du)=6(x^3-5x)^2
So (dy)/(dx)=6(3x^2-5)(x^3-5x)^2

Now adding both together,
(dy)/(dx)=2(2x+5)(x^2+5x)+6(3x^2-5)(x^3-5x)^2