Difference between revisions of "Derivative of tangent"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
| − | + | $$\dfrac{\mathrm{d}}{\mathrm{d}z} \tan(z) = \sec^2(z),$$ | |
| − | + | where $\tan$ denotes the [[tangent]] function and $\sec$ denotes the [[secant]] function. | |
| − | + | ||
| − | + | ==Proof== | |
| + | From the definition, | ||
| + | $$\tan(z) = \dfrac{\sin(z)}{\cos(z)},$$ | ||
| + | so using the [[quotient rule]], the [[derivative of sine]], the [[derivative of cosine]], the [[Pythagorean identity for sin and cos]], and the definition of [[secant]], | ||
| + | $$\dfrac{\mathrm{d}}{\mathrm{d}z} \tan(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \dfrac{\sin(z)}{\cos(z)} = \dfrac{\cos^2(z) + \sin^2(z)}{\cos^2(z)} = \dfrac{1}{\cos^2(z)} = \sec^2(z),$$ | ||
| + | as was to be shown. $\blacksquare$ | ||
| + | |||
| + | ==References== | ||
| + | *{{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Derivative of cosine|next=Derivative of cosecant}}: $4.3.107$ | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Proven]] | ||
Latest revision as of 00:35, 26 April 2017
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \tan(z) = \sec^2(z),$$ where $\tan$ denotes the tangent function and $\sec$ denotes the secant function.
Proof
From the definition, $$\tan(z) = \dfrac{\sin(z)}{\cos(z)},$$ so using the quotient rule, the derivative of sine, the derivative of cosine, the Pythagorean identity for sin and cos, and the definition of secant, $$\dfrac{\mathrm{d}}{\mathrm{d}z} \tan(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \dfrac{\sin(z)}{\cos(z)} = \dfrac{\cos^2(z) + \sin^2(z)}{\cos^2(z)} = \dfrac{1}{\cos^2(z)} = \sec^2(z),$$ as was to be shown. $\blacksquare$
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.3.107$
