Examples of Derivatives of Exponential Functions

Example: Differentiate ${a^{\sin x}} + {e^{\cos x}}$ with respect to $x$.

We have the 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 formula 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 as
$\begin{gathered} y = \ln \left( {\frac{{{e^x}}}{{1 + {e^x}}}} \right) \\ \Rightarrow y = \ln {e^x} – \ln \left( {1 + {e^x}} \right) \\ \end{gathered}$

Now differentiate both sides with respect to $x$
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}\ln {e^x} – \frac{d}{{dx}}\ln \left( {1 + {e^x}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{{e^x}}}\frac{d}{{dx}}{e^x} – \frac{1}{{1 + {e^x}}}\frac{d}{{dx}}\left( {1 + {e^x}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{{e^x}}}{{{e^x}}} – \frac{{{e^x}}}{{1 + {e^x}}} \\ \Rightarrow \frac{{dy}}{{dx}} = 1 – \frac{{{e^x}}}{{1 + {e^x}}} = \frac{{1 + {e^x} – {e^x}}}{{1 + {e^x}}} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{1 + {e^x}}} \\ \end{gathered}$