Question

How do you find the derivative of the function: `(arccos(x/6))^2`?

Answer 1

Let `h(x) = x/6`
Let `g(h) = arccos(h)`
Let `f(g) = g^2`
To differentiate, use the chain rule with a function nested within a function within a function.

Explanation:

The chain rule with a function nested within a function within a function:

`(d((arccos(x/6)^2)))/dx = (d(f(g(h(x)))))/dx`

`(d((arccos(x/6)^2)))/dx = (df)/(dg)(dg)/(dh)(dh)/dx" [1]"`

`(df)/(dg) = 2g = 2arccos(h) = 2arccos(x/6)`

`(dg)/(dh) = -1/sqrt(1 - h^2) = -1/sqrt(1 - (x/6)^2)`

`(dh)/dx = 1/6`

Substituting into equation [1]

`(d((arccos(x/6)^2)))/dx = (2arccos(x/6))(-1/sqrt(1 - (x/6)^2))(1/6)`

`(d((arccos(x/6)^2)))/dx = (-2arccos(x/6))/(6sqrt(1 - (x/6)^2))`

`(d((arccos(x/6)^2)))/dx = (-2arccos(x/6))/sqrt(36 - x^2)`