Question

How do you find the derivative of `tan^4 (x)`?

Answer 1

By using the power rule

`d/(dx)[(u(x))^n] = n [u(x)]^(n-1)`, where `u(x)` is a function of `x`,

and the chain rule

`d/(dx)[f(u)] = (df)/(du)(du)/(dx)`, where `f = f(u(x))`.

If we rewrite `tan^4(x)` as `(tanx)^4`, we have that:

• `f(u) = u^4`
• `u(x) = tanx`

As a result:

`color(blue)(d/(dx)[f(u)]) = (df)/(du)(du)/(dx)`

`d/(du)[u^4]cdot d/(dx)[tanx]`

`= 4u^3 cdot sec^2x`

`= color(blue)(4tan^3xsec^2x)`