What is the differential of (tanx) . Can you prove it please?

We can solve the differential of tan x using the first
principle, but an easier way to do it is to use the quotient rule as we know that tan x
= sin x / cos x.

[tan x]' = [sin x / cos
x]'

=> [(sin x)'*cos x - (sin x)*(cos x)']/(cos
x)^2

the derivative of sin x = cos x and that of cos x is -
sin x

=> [cos x* cos x  + sin x * sin x]/(cos
x)^2

=> [(cos x)^2 + (sin x)^2]/(cos
x)^2

=> 1 / (cos
x)^2

=> (sec
x)^2

The derivative of tan x is (sec
x)^2