Question

How do you differentiate `y=(x^4+3x^2-2)^5`?

Answer 1

`(dy)/(dx)=10x(2x^2+3)(x^4+3x^2-2)^4`

Explanation:

To differentiate `y=(x^4+3x^2-2)^5`, we use Chain Rule .

The concept under this rule is that of a function of a function, say `y, =f(g(x))`, where we have to find `(dy)/(dx)`. What we need to do is (a) to substitute `u=g(x)`, which gives us `y=f(u)`. Then we need to use a formula called Chain Rule, which states that `(dy)/(dx)=(dy)/(du)xx(du)/(dx)`.

Here we have `y=f(g*x))=g(x)^5` where `g(x)=x^4+3x^2-2`

Hence, `(dy)/(dx)=d/(d(x^4+3x^2-2))(x^4+3x^2-2)^5xxd/(dx)(x^4+3x^2-2)`

= `5(x^4+3x^2-2)^4xx(4x^3+6x)`

= `10x(2x^2+3)(x^4+3x^2-2)`