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JA5. Logarithmic Differentiation

Statement

Give one example of a mathematical idea from topics this week that you found creative and explain what you find creative about it. For example, you can choose an instance of creativity you experienced in your own problem-solving or something you witnessed in another person’s definition or reasoning.

Solution

Logarithmic differentiation is the creative instance that this text will discuss.

For functions of shape \(f(x) = x^n\) we can use the power rule to find the derivative. However, if we have a function of shape \(f(x) = g(x)^{h(x)}\) we cannot use the power rule to find the derivative as the exponent is not constant. Instead, we can use logarithmic differentiation to find the derivative (Herman & Strang, 2020, p.328).

The logarithmic differentiation is defined as follows:

\[ \begin{aligned} f(x) &= g(x)^{h(x)} \\ \ln(f(x)) &= \ln(g(x)^{h(x)}) \text{ ---(1)}\\ \ln(f(x)) &= h(x) \ln(g(x)) \text{ ---(2)} \\ \frac{f'(x)}{f(x)} &= h'(x) \ln(g(x)) + \frac{h(x) g'(x)}{g(x)} \text{ ---(3)}\\ f'(x) &= f(x) \left( h'(x) \ln(g(x)) + \frac{h(x) g'(x)}{g(x)} \right) \text{ ---(4)} \\ \end{aligned} \]

The creative about this method, that it is hard to think of, as the one is tempted to use the normal power rule in finding the derivative. To be surprised that if you integrate the results you will not get the original function back; then using the power rule was wrong.

Thus the use of logarithmic differentiation as it:

  • Ensures the correct derivative is found.
  • Simplifies the process of finding the derivative, despite adding some ln() functions which they should not cause any problems as their derivative is \(\frac{1}{x}\) and they have straightforward rules.
  • Reducing the complexity of the formula, as exponents always complex, and if they are of a variable or function nature, they are even more complex; thus, replacing the exponent with some multiplication and division is a good idea.

As an example, let’s try the exercise 349 from (Herman & Strang, 2020, p.331), which asks about the derivative of \(f(x) = x^{log_2{x}}\) using logarithmic differentiation:

\[ \begin{aligned} f(x) &= x^{log_2{x}} \\ \ln(f(x)) &= \ln(x^{log_2{x}}) \\ \ln(f(x)) &= log_2{x} \ln(x) \\ \frac{f'(x)}{f(x)} &= (log_2{x})' \ln(x) + \frac{log_2{x} \cdot (x)'}{x} \\ \frac{f'(x)}{f(x)} &= \frac{1}{x \ln(2)} \ln(x) + \frac{log_2{x}}{x} \\ f'(x) &= f(x) \left( \frac{1}{x \ln(2)} \ln(x) + \frac{log_2{x}}{x} \right) \\ f'(x) &= x^{log_2{x}} \left( \frac{1}{x \ln(2)} \ln(x) + \frac{log_2{x}}{x} \right) \\ f'(x) &= x^{log_2{x}} \left( \frac{\ln(x)}{x \ln(2)} + \frac{log_2{x}}{x} \right) \\ \end{aligned} \]

As we notice the final formula is not so pretty, but it would have been very complex to try to reach to it without using the logarithmic differentiation.

References

  • Herman, E. & Strang, G. (2020). Calculus volume 1. OpenStacks. Rice University.
  • The Organic Chemistry Tutor. (2016, October 14). Logarithmic differentiation Video. YouTube.