Derivative of Tangent Function

Theorem

 * $D_x \left({\tan x}\right) = \sec^2 x = \dfrac 1 {\cos^2 x}$

when $\cos x \ne 0$.

Proof
From the definition of the tangent function:
 * $\tan x = \dfrac {\sin x} {\cos x}$

From Derivative of Sine Function:
 * $D_x \left({\sin x}\right) = \cos x$

From Derivative of Cosine Function:
 * $D_x \left({\cos x}\right) = -\sin x$

Then:

This is valid only when $\cos x \ne 0$.

The result follows from the Secant is Reciprocal of Cosine.