# Examples of Derivatives of Logarithmic Functions

Example: Differentiate ${\log _{10}}\left( {\frac{{x + 1}}{x}} \right)$ with respect to $x$.

Consider the function $y = {\log _{10}}\left( {\frac{{x + 1}}{x}} \right)$

Differentiating both sides with respect to $x$, we have
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{1}{{\left( {\frac{{x + 1}}{x}} \right)\ln 10}}\frac{d}{{dx}}\left( {\frac{{x + 1}}{x}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{x}{{\left( {x + 1} \right)\ln 10}}\left[ {\frac{{x – \left( {x + 1} \right)}}{{{x^2}}}} \right] \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{ – 1}}{{x\left( {x + 1} \right)\ln 10}} \\ \end{gathered}$

Example: Find $\frac{{dy}}{{dx}}$, if the given function is $y = {x^{\cos y}}$

We have the given function
$y = {x^{\cos y}}$

Taking $\ln$ on both sides of the given function, we have
$\begin{gathered} \ln y = \ln {x^{\cos y}} \\ \Rightarrow \ln y = \cos y\ln x \\ \end{gathered}$

Differentiating both sides with respect to $x$, we have
$\begin{gathered} \frac{d}{{dx}}\ln y = \frac{d}{{dx}}\left( {\cos y\ln x} \right) \\ \Rightarrow \frac{1}{y}\frac{{dy}}{{dx}} = \cos y\frac{d}{{dx}}\left( {\ln x} \right) + \ln x\frac{d}{{dx}}\cos y \\ \Rightarrow \frac{1}{y}\frac{{dy}}{{dx}} = \cos y\frac{1}{x} – \ln x\sin y\frac{{dy}}{{dx}} \\ \Rightarrow \frac{1}{y}\frac{{dy}}{{dx}} + \sin y\frac{{dy}}{{dx}} = \cos y\frac{1}{x} \\ \Rightarrow \frac{{dy}}{{dx}}\left( {\frac{1}{y} + \sin y\ln x} \right) = \frac{{\cos y}}{x} \\ \Rightarrow \frac{{dy}}{{dx}}\left( {\frac{{1 + y\sin y\ln x}}{y}} \right) = \frac{{\cos y}}{x} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{y\cos y}}{{x\left( {1 + y\sin y\ln x} \right)}} \\ \end{gathered}$