Question

How do you differentiate `f(x) = (1/3) (arctan(3x))^2`?

Answer 1

`f'(x)=(2arctan(3x))/(1+9x^2)`

Explanation:

`"differentiate using the "color(blue)"chain rule"`

`"given "y=f(g(x))" then"`

`dy/dx=f'(g(x))xxg'(x)larrcolor(blue)"chain rule"`

`"noting that "d/dx(arctan(x))=1/(1+x^2)`

`f'(x)=1/3. 2(arctan(3x))xxd/dx(arctan(3x))`

`color(white)(f'(x))=2/3arctan(3x). 1/(1+(3x)^2)xxd/dx(3x)`

`color(white)(f'(x))=(2arctan(3x))/(1+9x^2)`