Find the derivation of each of the following from the first principle:

tan²x

Let f(x) = tan²x

We need to find the derivative of f(x) i.e. f’(x)

We know that,

…(i)

f(x) = tan²x

f(x + h) = tan²(x + h)

Putting values in (i), we get

Using:

[∵ sin A cos B – sin B cos A = sin(A – B)

& sin A cos B + sin B cos A = sin(A + B)]

Putting h = 0, we get

[∵ sin2x = 2sinxcosx]

= 2tanx sec²x

Hence, f’(x) = 2tanx sec²x