Derivative of secant

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Theorem

The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \sec(z)=\tan(z)\sec(z),$$ where $\sec$ denotes the secant and $\cot$ denotes the cotangent.

Proof

From the definition of secant, $$\sec(z) = \dfrac{1}{\cos(z)},$$ and so using the quotient rule, the derivative of cosine, and the definition of tangent, $$\dfrac{\mathrm{d}}{\mathrm{d}z} \sec(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \dfrac{1}{\cos(z)} = \dfrac{\sin(z)}{\cos^2(z)}=\tan(z)\sec(z),$$ as was to be shown.

References