Question

How do you find the derivative of `arctan(x^2y)`?

Answer 1

`d/dx(arctan(x^2y)) = (2xy)/(1 + (x^2y)^2)`

Explanation:

So, basically, you want to find `d/dx(arctan(x^2y))`.

We need to first observe that `y` and `x` have no relation to each other in the expression. This observation is very important, since now `y` can be treated as a constant with respect to `x`.

We first apply chain rule:
`d/dx(arctan(x^2y)) = d/(d(x^2y))(arctan(x^2y)) xx d/dx(x^2y) = 1/(1 + (x^2y)^2) xx d/dx(x^2y)`.

Here, as we mentioned earlier, `y` is a constant with respect to `x`. So,

`d/dx(x^2 color(red)(y)) = color(red)(y) xx d/dx(x^2) = 2xy`

So, `d/dx(arctan(x^2y)) = 1/(1 + (x^2y)^2) xx 2xy = (2xy)/(1 + (x^2y)^2)`