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Derivative of Tan Demonstration

Tan

Demonstration of derivative of Tan

This demonstration is a simple application of the quotient rule to the definition of Tan.

{$\color{crimson}{y = \tan(x)}$} and {$\color{blue}{y=d/dx \tan(x) = \sec^2 (x)}$}

{\large \begin{align} {d \over {d\theta}} \tan\theta &= {d \over {d\theta}} \left( {{\sin\theta} \over {\cos\theta}} \right) \cr &= {{ \cos\theta{d \over {d\theta}}\sin\theta -\sin\theta {d \over {d\theta}}\cos\theta} \over {\cos^2\theta}} \tag{quotient rule} \cr &= {{\cos^2\theta} \over {\cos^2\theta}} + {{\sin^2\theta} \over {\cos^2\theta}} \cr &= 1 + \tan^2\theta \end{align}}

{$$\Large \therefore {d \over {d\theta}} \tan\theta = \sec^2\theta \tag{Pythagorean identity}$$}

Sources:

1. FooPlot: Online graphing calculator and function plotter

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Categories: Derivative Demonstrations

This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.