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}