How do we differentiate y = arctan(x)?

Step 1: Rearrange y = arctan(x) as tan(y) = x.

Step 2: Use implicit differentiation to differentiate this with respect to x, which gives us:

(dy/dx)*(sec(y))^2 = 1.

Step 3: Rearrange this equation to give us:

dy/dx = 1/(sec(y))^2.

Step 4: Use a trigonometric identity to substitute and find that:

dy/dx = 1/(1+((tan(y))^2).

Step 5: Recall that x = tan(y) and substitute this to find: 

dy/dx = 1/(1+x^2).

Done.