Question

What is the derivative of $y = 2{\tan ^{ - 1}}\left( {{e^x}} \right)$ with respect to $x$.

Answer 1

Hint: In the given problem, we are required to differentiate $y = 2{\tan ^{ - 1}}\left( {{e^x}} \right)$ with respect to x. Since, $y = 2{\tan ^{ - 1}}\left( {{e^x}} \right)$ is a composite function. So, differentiation of $y = 2{\tan ^{ - 1}}\left( {{e^x}} \right)$ with respect to x will be done layer by layer using the chain rule of differentiation. Also the derivative of ${\tan ^{ - 1}}(x)$with respect to $x$ must be remembered.

Complete step by step answer:
Now, $\dfrac{d}{{dx}}\left( {2{{\tan }^{ - 1}}\left( {{e^x}} \right)} \right)$. Taking the constant outside the differentiation in order to apply the chain rule of differentiation. So, we get,
$2\dfrac{d}{{dx}}\left( {{{\tan }^{ - 1}}\left( {{e^x}} \right)} \right)$
Now, Let us assume $u = {e^x}$. So substituting ${e^x}$ as $u$, we get,
$2\dfrac{d}{{dx}}\left( {{{\tan }^{ - 1}}\left( u \right)} \right)$
Now ,we know that derivative of inverse tangent function ${\tan ^{ - 1}}\left( x \right)$ with respect to x is $\left( {\dfrac{1}{{1 + {x^2}}}} \right)$. So, we get $\left( {\dfrac{2}{{1 + {u^2}}}} \right)\left( {\dfrac{{du}}{{dx}}} \right)$.

Now, putting back $u$as ${e^x}$, we get,
\[\left( {\dfrac{2}{{1 + {{\left( {{e^x}} \right)}^2}}}} \right)\left( {\dfrac{{d\left( {{e^x}} \right)}}{{dx}}} \right)\] because \[\dfrac{{du}}{{dx}} = \dfrac{{d({e^x})}}{{dx}}\]
Now, we know that the derivative of exponential function \[{e^x}\] with respect to x is \[{e^x}\]. So, we get,
\[\left( {\dfrac{2}{{1 + {e^{2x}}}}} \right)\left( {{e^x}} \right)\]
Simplifying the expression, we get,
\[\therefore \dfrac{{2{e^x}}}{{1 + {e^{2x}}}}\]

Therefore, the derivative of $y = 2{\tan ^{ - 1}}\left( {{e^x}} \right)$ with respect to x is \[\left( {\dfrac{{2{e^x}}}{{1 + {e^{2x}}}}} \right)\].

Note: The derivatives of basic trigonometric functions must be learned by heart in order to find derivatives of complex composite functions using chain rule of differentiation. The chain rule of differentiation involves differentiating a composite by introducing new unknowns to ease the process and examine the behaviour of function layer by layer and hence, the derivative of a composite function can be found using the same rule.