Derivatives Of Logarithmic Functions

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Source: http://www.phengkimving.com/calc_of_one_real_var/07_the_exp_and_log_func/07_03_gen_exp_and_log_func_files/image086.gif

This article is about finding derivatives of logarithmic functions. This topic is typically found in introductory calculus courses.


Table of Contents

  1. Brief Review of Exponential Functions and Logarithmic Functions
  2. Derivatives of Logarithmic Functions – Formulas
  3. Examples


Brief Review of Exponential Functions and Logarithmic Functions

An exponential function is a function of the form:

\displaystyle a^{x}

where a is a non-zero number and x is a variable.

The logarithmic function is the inverse function of the exponential function. The logarithmic function is of the form:

\displaystyle \log_a (x)

where the base a is a non-zero number and x is a variable.

If we have the exponential function of \text{e}^{x} with Euler’s constant of e \approx 2.71828 then the corresponding inverse would be \log_e(x) = \ln(x)}.


Derivatives of Logarithmic Functions – Formulas

Given that f(x) is of the form f(x) = \log_a(g(x)) then the derivative f'(x) is:

\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{1}{\text{ln} (a)} \cdot \dfrac{1}{g(x)} \cdot \dfrac{d}{dx} g(x) \\ & = & \dfrac{1}{\text{ln} (a)} \cdot \dfrac{1}{g(x)} \cdot g'(x) \\ \end{array} \\

A more simpler case is when we have f(x) = \log_a (x). The derivative would be:

\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{1}{\text{ln} (a)} \cdot \dfrac{1}{x} \cdot \dfrac{d}{dx} x \\ & = & \dfrac{1}{\text{ln} (a)} \cdot \dfrac{1}{x} \cdot 1 \\ & = & \dfrac{1}{\ln (a)} \cdot \dfrac{1}{g(x)} \\ \end{array} \\

The most simplest and most common case is taking the derivative of f(x) = \ln(x).

\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{d}{dx} \ln(x) \\ & = & \dfrac{1}{\ln(e)} \cdot \dfrac{1}{x} \cdot \dfrac{d}{dx} x\\ & = & \dfrac{1}{1} \cdot \dfrac{1}{x} \cdot 1\\ & = & \dfrac{1}{x} \\ \end{array} \\

(Note that ln(e) = 1.)


Examples

Here are some examples of differentiating logarithmic functions.

Example One:

Given that f(x) = \ln(2x) then the derivative is:

\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{d}{dx} \ln(2x) \\ & = & \dfrac{1}{\ln(e)} \cdot \dfrac{1}{2x} \cdot \dfrac{d}{dx} 2x\\ & = & \dfrac{1}{1} \cdot \dfrac{1}{2x} \cdot 2\\ & = & \dfrac{1}{x} \\ \end{array} \\

Example Two:

If we have f(x) = \log_5(x) then the derivative is:

\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{1}{\text{ln}(5) \cdot x} \cdot 1 \\ & = & \dfrac{1}{\ln(5) \cdot x} \\ \end{array} \\

Example Three:

The derivative of the function f(x) = \log_{10}(x^2) is:

\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{1}{\ln(10) \cdot x^2} \cdot 2x \\ & = & \dfrac{2}{\ln(10) \cdot x} \\ \end{array} \\

Example Four:

This last example is more involved. Suppose that the function g(x) is g(x) = \text{ln}(\text{ln}(x^5)). At first, you may be terrified. However, if you look at this more closely this is a case of using chain rule. You would need to differentiate multiple times. The derivative of g is as follows.

\displaystyle \begin{array} {lcl} g'(x) & = & \dfrac{d}{dx} \text{ln}(\text{ln}(x^5)) \\ & = & \dfrac{1}{\text{ln}(x^5)} \cdot \dfrac{d}{dx} \text{ln}(x^5) \\ & = & \dfrac{1}{\text{ln}(x^5)} \cdot \dfrac{1}{x^5} \cdot \dfrac{d}{dx} (x^5) \\ & = & \dfrac{1}{\text{ln}(x^5)} \cdot \dfrac{1}{x^5} \cdot 5x^4 \\ & = & \dfrac{5}{\text{ln}(x^5) \cdot x}\\ \end{array} \\

When dealing with chain rule cases, go from the outside to the inside. From the first line to the second line the derivative of \ln( \cdot ) is \dfrac{1}{\ln( \cdot )}. Then we take the derivative of the inside which is ( \cdot ).

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