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.
SC
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