Question

How do you differentiate `f(x) = sqrt(arctan(3x)` using the chain rule?

Answer 1

Chain rule says that `d/dxf[g(x)]=f'[g(x)]g'(x)`.
Here we have three functions that are `3x`, `arctan` and `sqrt`.
We start from the most "external" that is the square root
We know that

`d/dxsqrt(x)=1/(2sqrt(x))`

then, applying the chain rule we have

`d/dxsqrt(arctan(3x))=1/(2sqrt(arctan(3x)))d/dxarctan(3x)`.

We have now to calculate the derivative of `arctan(3x)`

We know that

`d/dxarctan(x)=1/(1+x^2)`

then we reapply the chain rule

`d/dxarctan(3x)=1/(1+(3x)^2)d/dx3x`

and finally the easy one

`d/dx3x=3`.

We substitute everything back and write

`d/dxsqrt(arctan(3x))=3/(2sqrt(arctan(3x))(1+9x^2)`