Question

How to find the derivative of `x^4`?

Answer 1

`(d color(red)(x^4))/(dx) = color(blue)(4x^3)`

Explanation:

In general
`color(white)("XXX")(d color(red)(ax^b))/(dx) = color(blue)(b*ax^(b-1))`

This is a standard formula, but you could derive it from basic definitions of derivative:
`color(white)("XXX")(df(x))/dx = lim_(hrarr0)(f(x+h)-f(x))/h`

In this case:
`color(white)("XXX")(dx^4)/(dx) = lim_(hrarr0) ((x+h)^4-x^4)/h`

`color(white)("XXXXX")=lim_(hrarr0)(cancel(x^4)+4x^3h+6x^2h^2+4xh^3+h^4 cancel(- x^4))/h`

`color(white)("XXXXX")=lim_(hrarr0)4x^3+6x^2h+4xh^2+h^3`

`color(white)("XXXXX")=4x^3`