Examples of Derivatives of Exponential Functions

Example: Differentiate {a^{\sin x}} + {e^{\cos x}} with respect to x.
We have given function

y = {a^{\sin x}} + {e^{\cos x}}


Now differentiate both sides with respect to x

\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( {{a^{\sin x}} + {e^{\cos x}}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{d}{{dx}}{a^{\sin x}} + \frac{d}{{dx}}{e^{\cos x}} \\ \end{gathered}


Using the derivative formulae for exponential functions, we have

\begin{gathered} \Rightarrow \frac{{dy}}{{dx}} = {a^{\sin x}}\ln a\frac{d}{{dx}}\sin x + {e^{\cos x}}\frac{d}{{dx}}\cos x \\ \Rightarrow \frac{{dy}}{{dx}} = {a^{\sin x}}\ln a\cos x - {e^{\cos x}}\sin x \\ \end{gathered}


Example: Find \frac{{dy}}{{dx}}, if the given function is

y = \ln \left( {\frac{{{e^x}}}{{1 + {e^x}}}} \right)


We have the given function

\begin{gathered} y = \ln \left( {\frac{{{e^x}}}{{1 + {e^x}}}} \right) \\ \Rightarrow y = \ln {e^x} - \ln \left( {1 + {e^x}} \right) \\ \end{gathered}


Now differentiate both sides with respect to x

\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}\ln {e^x} - \frac{d}{{dx}}\ln \left( {1 + {e^x}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{{e^x}}}\frac{d}{{dx}}{e^x} - \frac{1}{{1 + {e^x}}}\frac{d}{{dx}}\left( {1 + {e^x}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{{e^x}}}{{{e^x}}} - \frac{{{e^x}}}{{1 + {e^x}}} \\ \Rightarrow \frac{{dy}}{{dx}} = 1 - \frac{{{e^x}}}{{1 + {e^x}}} = \frac{{1 + {e^x} - {e^x}}}{{1 + {e^x}}} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{1 + {e^x}}} \\ \end{gathered}

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