`f(x) = arctan(e^x)` Find the derivative of the function

Wednesday, February 3, 2016

$\displaystyle{f{{\left({x}\right)}}}={\arctan{{\left({{e}}^{{x}}\right)}}}$ Find the derivative of the function

The given function: $\displaystyle{f{{\left({x}\right)}}}={\arctan{{\left({{e}}^{{x}}\right)}}}$ is in a form of inverse trigonometric function.

It can be evaluated using the derivative formula for inverse of tangent function:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\arctan{{\left({u}\right)}}}=\frac{{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}}{{{1}+{{x}}^{{2}}}}$ .

We let $\displaystyle{u}={{e}}^{{x}}$ then $\displaystyle\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}=\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({{e}}^{{x}}\right)}={{e}}^{{x}}$ .

Applying the the formula, we get:

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{d}}{{{\left.{d}{x}\right.}}}{\arctan{{\left({{e}}^{{x}}\right)}}}$

$\displaystyle=\frac{{{e}}^{{x}}}{{{1}+{{\left({{e}}^{{x}}\right)}}^{{2}}}}$

Using the law of exponent: $\displaystyle{{\left({{x}}^{{n}}\right)}}^{{m}}={{x}}^{{{n}\cdot{m}}}$ , we may simplify the part:

$\displaystyle{{\left({{e}}^{{x}}\right)}}^{{2}}={{e}}^{{{\left({x}\cdot{2}\right)}}}={{e}}^{{{2}{x}}}$

The derivative of the function $\displaystyle{f{{\left({x}\right)}}}={\arctan{{\left({{e}}^{{x}}\right)}}}$ becomes:

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{{e}}^{{x}}}{{{1}+{{e}}^{{{2}{x}}}}}$ as the Final Answer.

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Wednesday, February 3, 2016

$\displaystyle{f{{\left({x}\right)}}}={\arctan{{\left({{e}}^{{x}}\right)}}}$ Find the derivative of the function

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Wednesday, February 3, 2016

Find the derivative of the function

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Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{f{{\left({x}\right)}}}={\arctan{{\left({{e}}^{{x}}\right)}}}$ Find the derivative of the function

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?