Difference between revisions of "Derivative of tangent"

From specialfunctionswiki
Jump to: navigation, search
(Proof)
 
(5 intermediate revisions by the same user not shown)
Line 1: Line 1:
<div class="toccolours mw-collapsible mw-collapsed">
+
==Theorem==
<strong>[[Derivative of tangent|Proposition]]:</strong> $\dfrac{d}{dx}$[[Tangent|$\tan$]]$(x)=$[[Secant|$\sec$]]$^2(x)$
+
The following formula holds:
<div class="mw-collapsible-content">
+
$$\dfrac{\mathrm{d}}{\mathrm{d}z} \tan(z) = \sec^2(z),$$
<strong>Proof:</strong> █
+
where $\tan$ denotes the [[tangent]] function and $\sec$ denotes the [[secant]] function.
</div>
+
 
</div>
+
==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