Derivative of the Real Tangent Function

Theorem: Derivative of the Real Tangent Function

The real tangent function is differentiable (everywhere it is defined) and its derivative is the recirpocal of the square of the real cosine function:

$$(\tan x{)}^{\prime }=\frac{1}{{\cos }^{2}x}=1+{\tan }^{2}x$$

PROOF

$$(\tan x{)}^{\prime }={\left(\frac{\sin x}{\cos x}\right)}^{\prime }=\frac{(\sin x{)}^{\prime }\cos x-(\cos x{)}^{\prime }\sin x}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}$$