If f (x) = 4arctan (7x), find f' (x). Find f' (4).

f' (x) = (4 arctan (7x)) '

f' (x) = 4 (arctan (7x)) '

By the chain rule,

f' (x) = 4 / (1 + (7x) ^2) * (7x) '

f' (x) = 28 / (1 + 49x^2)

and hence

f' (4) = 28 / (1 + 49*16) = 28/785

In case you're not sure about the derivative of arctan: If y = arctan (x), then x = tan (y). Differentiating both sides with respect to x gives

1 = sec^2y y' = (1 + tan^2y) y' = (1 + x^2) y'

==> y' = 1 / (1 + x^2)