Differentiate the following:?

Question

tan^-1(x)

The choices are:
a) 1 / sqrt(1 - x^2)
b) 1 / sqrt(x^2 - 1)
c) 1 / (1 + x^2)
d) 1 / (1 - x^2)

Answer 1

The answer is C:

Proof:

tan﴾tan ‾ ¹(x)﴿ = x for all x

let y = tan ‾ ¹(x)

so we have:

tan(y) = x

now differentiate both sides (you need to use implicit differentiation)

sec²(y)dy/dx = 1

dy/dx = 1/ sec²(y)

by trig. Identity:

dy/dx = 1/ ﴾ 1 + tan²(y) ﴿

But look above, y = tan ‾ ¹(x), so tan²(y) = tan²( tan ‾ ¹(x)) = x²

Thus

dy/dx = 1/ ﴾ 1 + x²﴿

QED

Answer 2

d