`y = e^(2x) tan(2x)` Find the derivative.

Find the derivative if $\displaystyle{y}={{e}}^{{{2}{x}}}{\tan{{\left({2}{x}\right)}}}$ :

Let u=2x and rewrite as:

$\displaystyle{y}={{e}}^{{u}}{\tan{{u}}}$

Use the product rule noting that $\displaystyle\frac{{d}}{{{d}{u}}}{{e}}^{{u}}={{e}}^{{u}}{d}{u},\frac{{d}}{{{d}{u}}}{\tan{{u}}}={{\sec}}^{{2}}{u}{d}{u}$ :

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{d}{u}}}={{e}}^{{u}}{d}{u}{\tan{{u}}}+{{e}}^{{u}}{{\sec}}^{{2}}{u}{d}{u}$

Since u=2x, du=2 so we can rewrite as:

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={2}{{e}}^{{{2}{x}}}{\tan{{\left({2}{x}\right)}}}+{2}{{e}}^{{{2}{x}}}{{\sec}}^{{{2}}}{2}{x}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={2}{{e}}^{{{2}{x}}}{\left({\tan{{\left({2}{x}\right)}}}+{{\sec}}^{{2}}{\left({2}{x}\right)}\right)}$

Ninenotes

Sunday, August 31, 2014

$\displaystyle{y}={{e}}^{{{2}{x}}}{\tan{{\left({2}{x}\right)}}}$ Find the derivative.

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