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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Derivatives of Logarithmic Functions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 18:49:55 1997}
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\section{Derivatives of Logarithmic Functions\label%
{Derivative of logarithmic function}}

\label{Derivative of natural logarithm}The logarithmic function $y=\ln x$ is
the inverse of the differentiable function $y=e^{x}$. Differentiating 
\[
e^{y}=x 
\]
implicitly with respect to $x$ yields 
\begin{eqnarray*}
e^{y}\,\,\dfrac{dy}{dx} &=&1 \\
\dfrac{dy}{dx} &=&\dfrac{1}{e^{y}}=\dfrac{1}{x}
\end{eqnarray*}
Thus 
\[
\dfrac{d}{dx}(\ln \,x)=\dfrac{1}{x} 
\]

\subsubsection{\label{Derivative of general logarithmic function}General
Logarithmic Function}

A logarithmic function with base $a$ can be expressed in terms of the
natural logarithmic function: 
\[
\log _{a}\,x=\frac{\ln \,x}{\ln \,a} 
\]
Since $\ln a$ is a constant, you can differentiate as follows: 
\[
\frac{d}{dx}\,\,\log _{a}\,x=\frac{d}{dx}\,\,\frac{\ln \,x}{\ln \,a}=\frac{1%
}{\ln \,a}\,\,\frac{d}{dx}\,\,\ln \,x=\frac{1}{x\,\ln \,a} 
\]
Thus 
\[
\text{ }\dfrac{d}{dx}\,\,\log _{a}\,x=\dfrac{1}{x\ln a} 
\]

\subsubsection{\label{Derivative of general exponential function}General
Exponential Function}

\begin{theorem}
For $a>0$, 
\[
\dfrac{d}{dx}\,\,a^{x}=a^{x}\,\ln \,a 
\]
\end{theorem}

\subsubsection{\QTR{BOXHEAD}{Logarithmic Differentiation \label%
{Logarithmic differentiation}}}

The calculation of derivatives of complicated functions involving products,
quotients, or powers can often be simplified by taking logarithms. This
method is called \textbf{logarithmic differentiation.}

\begin{enumerate}
\item[1.]  Take logarithms of both sides of an equation $y=f(x)$.

\item[2.]  Differentiate implicitly with respect to $x$.

\item[3.]  Solve the resulting equation for $y^{\prime }$.
\end{enumerate}

\begin{example}
To differentiate $y=\dfrac{\left( x^{2}-3x\right) \sin x}{x^{2}+3x+1}$

\begin{enumerate}
\item  $\ln y=\ln \left( \dfrac{\left( x^{2}-3x\right) \sin x}{x^{2}+3x+1}%
\right) =\allowbreak \ln x+\ln \left( x-3\right) +\ln \left( \sin x\right)
-\ln \left( x^{2}+3x+1\right) $

\item  $\frac{1}{y}y^{\prime }=\frac{1}{x}+\frac{1}{x-3}+\dfrac{1}{\sin x}%
\cos x-\dfrac{1}{x^{2}+3x+1}\left( 2x+3\right) $

\item  $y^{\prime }=y\left( \dfrac{1}{x}+\dfrac{1}{x-3}+\dfrac{\cos x}{\sin x%
}-\dfrac{\left( 2x+3\right) }{x^{2}+3x+1}\right) \allowbreak $
\end{enumerate}
\end{example}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Computing a derivative in Scientific Notebook}{}{}{%
../../Help/DM7-3.tex#Derivative}}

{\small \hyperref{Logarithmic functions }{}{}{../../Help/DM3-5.tex#Logarithms%
}}

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