Derivative of Tangent Function/Proof 1

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

From Derivative of Sine Function:
 * $\map {\dfrac \d {\d x} } {\sin x} = \cos x$

From Derivative of Cosine Function:
 * $\map {\dfrac \d {\d x} } {\cos x} = -\sin x$

Then:

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

The result follows from the Secant is Reciprocal of Cosine.

Proof

 * : $\text {II}$. Calculus: Differentiation: Quotient