Find the derivative of the function.

*g*(

*x*) = (1 + 4

*x*)5(4 +

*x*−

*x*2)7

we just started using the chain rule today. so its only the basics.

### 1 Answer | Add Yours

I think you have mentioned,

`g(x) = (1+4x)^5(4+x-x^2)^7`

Let `u = (1+4x)^5` and `v = (4+x-x^2)^7`

Then according to product rule,

`g'(x) = u'v+uv'`

`u = (1+4x)^5`

Let,

`1+4x = t`

`(dt)/(dx) = 4`

Then,

`u = t^5`

Then using chain rule,

`(du)/(dx) = (d(t^5))/(dt) xx (dt)/(dx)`

`(du)/(dx) = 5t^4 xx 4`

`(du)/(dx) = 20t^4`

`(du)/(dx) = 20(1+4x)^4`

`v = (4+x-x^2)^7`

Let `s = 4+x-x^2`

`(ds)/(dx) = 1-2x`

`(dv)/(dx) = (d(s^7))/(ds) xx (ds)/(dx)`

`(dv)/(dx) = 7s^6 xx (1-2x)`

`(dv)/(dx) = 7(4+x-x^2)^6(1-2x)`

Therefore,

`g'(x) = 20(1+4x)^4 xx (4+x-x^2)^7 + (1+4x)^5 xx 7(4+x-x^2)^6(1-2x)`

`g'(x) = (1+4x)^4 (4+x-x^2)^6 (20(4+x-x^2)+42(1+4x)(1-2x))`

`g'(x) = 2(1+4x)^4 (4+x-x^2)^6(10(4+x-x^2)+21(1+2x-8x^2))`

`g'(x) = 2(1+4x)^4 (4+x-x^2)^6(40+10x-10x^2+21+42x-168x^2)`

`g'(x) = 2(1+4x)^4 (4+x-x^2)^6(61+52x-178x^2)`

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