Question

Please help me I have mistake​

Answer 1

`y = (2ax^2 + c)^2 (bx^2 - cx)^(-1)`

By the product rule,

`(dy)/(dx) = (d(2ax^2+c)^2)/(dx) (bx^2-cx)^(-1) + (2ax^2+c)^2 (d(bx^2-cx)^(-1))/(dx)`

For the first derivative on the right side, let
`u=2ax^2+c`. Then by the chain rule,

`(d(2ax^2+c)^2)/(dx) = (du^2)/(dx) \\\\ ~~~~~~~~~~~~~~~~~~ = (du^2)/(du)(du)/(dx) \\\\ ~~~~~~~~~~~~~~~~~~ = 2u (du)/(dx) \\\\ ~~~~~~~~~~~~~~~~~~ = 2(2ax^2+c) (d(2ax^2+c))/(dx)`

and by the power rule, this reduces to

`(d(2ax^2+c)^2)/(dx) = 2 (2ax^2+c) (4ax) \\\\ ~~~~~~~~~~~~~~~~~~ = 8ax (2ax^2 + c)`

For the other derivative, let
`v = bx^2-cx`. Then

`(d(bx^2-cx)^(-1))/(dx) = (dv^(-1))/(dx) \\\\ ~~~~~~~~~~~~~~~~~~~~ = (dv^(-1))/(dv)(dv)/(dx) \\\\ ~~~~~~~~~~~~~~~~~~~~ = -v^(-2) (dv)/(dx) \\\\ ~~~~~~~~~~~~~~~~~~~~ = -(bx^2-cx)^(-2)(d(bx^2-cx))/(dx)`

which reduces to

`(d(bx^2-cx)^(-1))/(dx) = -(bx^2 - cx)^(-2) (2bx - c)`

So one expression for the derivative we want is

`(dy)/(dx) = 8ax (2ax^2 + c) (bx^2-cx)^(-1) - (2ax^2+c)^2 (bx^2 - cx)^(-2) (2bx - c)`

Answer 2

See below

Explanation:

Here is one possible mistake:

Let y= u = ( 2ax^2 +c)^2 expand

4 a^2 x^4 + 4 a x^2 c + c^2

dy/ dx then becomes

16 a^2 x^3 + 8 a x c <====== you have 8 a^2x^3 + 8axc

MAYBE ??