Question

How do you find the derivative of `(1+4x)^5(3+x-x^2)^8` using the chain rule?

Answer 1

`y'=8(1-2x)(1+4x)^5(3+x-x^2)^7 +20(1+4x)^4(3+x-x^2)^8`

Explanation:

`y=(1+4x)^5 (3+x-x^2)^8`

Use product rule `(fg)'=fg'+gf'` and chain rule `(f(g(x))'=f'(g(x))*g'(x)`

`f=(1+4x)^5,` `g=(3+x-x^2)^8`

`f'=5(1+4x)^4 * 4,` `g'=8(3+x-x^2)^7 * (1-2x)`

`y'=fg'+gf'`

`y'=8(1-2x)(1+4x)^5(3+x-x^2)^7 +20(1+4x)^4(3+x-x^2)^8`