Question

`({ {x}^(2) - 4})^(5) ( {4x - 5})^(4)`
can someone help me differentiate? ​

Answer 1

Let
`u=x^2-4` and
`v=4x-5`. By the product rule,

`(\mathrm d(u^5v^4))/(\mathrm dx)=(\mathrm d(u^5))/(\mathrm dx)v^4+u^5(\mathrm d(v^4))/(\mathrm dx)`

By the power rule, we have
`(u^5)'=5u^4` and
`(v^4)'=4v^3`, but
`u,v` are functions of
`x`, so we also need to apply the chain rule:

`(\mathrm d(u^5))/(\mathrm dx)=5u^4(\mathrm du)/(\mathrm dx)`

`(\mathrm d(v^4))/(\mathrm dx)=4v^3(\mathrm dv)/(\mathrm dx)`

and we have

`(\mathrm du)/(\mathrm dx)=2x`

`(\mathrm dv)/(\mathrm dx)=4`

So we end up with

`(\mathrm d(u^5v^4))/(\mathrm dx)=10xu^4v^4+16u^5v^3`

Replace
`u,v` to get everything in terms of
`x`:

`(\mathrm d((x^2-4)^5(4x-5)^4))/(\mathrm dx)=10x(x^2-4)^4(4x-5)^4+16(x^2-4)^5(4x-5)^3`

We can simplify this by factoring:

`10x(x^2-4)^4(4x-5)^4+16(x^2-4)^5(4x-5)^3=2(x^2-4)^4(4x-5)^3\bigg(5x(4x-5)+8(x^2-4)\bigg)`

`=2(x^2-4)^4(4x-5)^3(28x^2-57)`