Differentiation, derivatives and Taylor approximations: Differentiating exponential and logarithmic functions

Theory Derivatives of logarithmic functions

Now that we know the derivative of an exponential function we can also determine the derivative of a logarithmic function. To remember:

Derivative of a logarithmic function If \(f(x)=\log_a(x)\) then \(\displaystyle f'(x)=\frac{1}{x\cdot\ln(a)}\) for any base \(a>0\) , \(a\neq 1\).

For math enthusiasts we present the proof of \(\displaystyle\bigl(\log_a(x)\bigr)'=\frac{1}{x\cdot\ln(a)}\) .

Note that the identical function \(s(x)=x\) is equal to the composition of the function \(f(u)=a^u\) with \(u=\log_{a}(x)\). According to the chain rule and the formula for the derivative of a power function the following is valid: \[1=s'(x)=f'\bigl(u(x)\bigr)\cdot u'(x)= a^{u}\cdot \ln(a)\cdot u'(x)=x\cdot \ln(a)\cdot u'(x)\] So \[u'(x)=\frac{1}{x\cdot \ln(a)}\]

Calculate the derivative of \(f(x)=\log_{19}(8x)\)
\(f(x)=\log_{19}(8x)\) is the composition of \(y=\log_{19}(u)\) and \(u=8x\).
Then: \(\displaystyle \frac{\dd y}{\dd u}=\frac{1}{\ln(19)\cdot u}\) and \(\displaystyle \frac{\dd u}{\dd x}=8\). The chain rule yields: \[\begin{aligned}f'(x)&=\frac{\dd y}{\dd u}\cdot \frac{\dd u}{\dd x}\\[0.25cm] &=\frac{1}{\ln(19)\cdot 8x}\cdot 8\\[0.25cm] &=\frac{1}{\ln(19)}\cdot \frac{1}{x}\end{aligned}\] Note that the function \(f(x)\) has the following property: \(f'(x)\) is a multiple of \(\displaystyle\frac{1}{x}\) .
New example

The \(\mathrm{pH}\) as a measure of acidity is defined as \[\mathrm{pH}=-\log_{10}\biggl(\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]\biggr)\] If you consider \(\mathrm{pH}\) as a function of the concentration \(\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]\), then the instantaneous change of \(\mathrm{pH}\) is given by \[\mathrm{pH}'=\frac{\dd\,\mathrm{pH}}{\dd\,\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]}=-\frac{1}{\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]\cdot\ln(10)}\approx -\frac{0.434}{\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]}\]

A particular case is the derivative of the natural logarithm:

Derivative of ln If \(f(x)=\ln(x)\) then \(\displaystyle f'(x)=\frac{1}{x}\).

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