Question

What is the derivative of `arctan(x/3)`?

Answer 1

`1/(3+x^2/3)`

Explanation:

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

`•color(white)(x)d/dx(arctan(f(x)))=1/(1+(f(x))^2)xxf'(x)`

`rArrd/dx(arctan(x/3))`

`=1/(1+x^2/9)xxd/dx(1/3x)`

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

Answer 2

If you haven't memorized the drivative of `arctan(x)`. (or you don't trust your memory), see below.

Explanation:

`y = arctan(x/3)`

`tany = x/3`

Diiferentiate implicitly.

`sec^2 y dy/dx = 1/3`

`dy/dx = 1/3 cos^2(y)`

Use `tany = x/3` to find `cos y = 3/sqrt(x^2+9)` (See Note below)

`dy/dx = 1/3 (3/sqrt(x^2+9))^2`

`dy/dx = 3/(x^2+9)`

Note
There are several possible methods to do this.
I like to draw a right triangle with one angle labeled `y`. The side opposite `y` is `x` and the adjacent side is `3`, so ythe hypotenuse is `sqty(x^2+9)` and the cosine is `3/sqrt(x^2+9)`
Others prefer to use `tan^2 y + 1 = sec^2 y` to find `sec^2y` and the invert.