Difference between revisions of "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. $\blacksquare$ | ||
| + | |||
| + | ==References== | ||
| + | *{{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Derivative of cosecant|next=Derivative of cotangent}}: $4.3.109$ | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Proven]] | ||
Latest revision as of 00:34, 26 April 2017
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. $\blacksquare$
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.3.109$
