Logarithmic Differentiation

Learning Outcomes

  • Use logarithmic differentiation to determine the derivative of a function

At this point, we can take derivatives of functions of the form [latex]y=(g(x))^n[/latex] for certain values of [latex]n[/latex], as well as functions of the form [latex]y=b^{g(x)}[/latex], where [latex]b>0[/latex] and [latex]b\ne 1[/latex]. Unfortunately, we still do not know the derivatives of functions such as [latex]y=x^x[/latex] or [latex]y=x^{\pi}[/latex]. These functions require a technique called logarithmic differentiation, which allows us to differentiate any function of the form [latex]h(x)=g(x)^{f(x)}[/latex]. It can also be used to convert a very complex differentiation problem into a simpler one, such as finding the derivative of [latex]y=\frac{x\sqrt{2x+1}}{e^x \sin^3 x}[/latex]. We outline this technique in the following problem-solving strategy.

Problem-Solving Strategy: Using Logarithmic Differentiation

  1. To differentiate [latex]y=h(x)[/latex] using logarithmic differentiation, take the natural logarithm of both sides of the equation to obtain [latex]\ln y=\ln (h(x))[/latex].
  2. Use properties of logarithms to expand [latex]\ln (h(x))[/latex] as much as possible.
  3. Differentiate both sides of the equation. On the left we will have [latex]\frac{1}{y}\frac{dy}{dx}[/latex].
  4. Multiply both sides of the equation by [latex]y[/latex] to solve for [latex]\frac{dy}{dx}[/latex].
  5. Replace [latex]y[/latex] by [latex]h(x)[/latex].

It may be useful to review your properties of logarithms. These will help us in step 2 to expand our logarithmic function.

Recall: Properties of logarithms

The Product Rule for Logarithms [latex]{\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)[/latex]
The Quotient Rule for Logarithms [latex]{\mathrm{log}}_{b}\left(\frac{M}{N}\right)={\mathrm{log}}_{b}M-{\mathrm{log}}_{b}N[/latex]
The Power Rule for Logarithms [latex]{\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M[/latex]
The Change-of-Base Formula [latex]{\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\text{ }n>0,n\ne 1,b\ne 1[/latex]

Example: Using Logarithmic Differentiation

Find the derivative of [latex]y=(2x^4+1)^{\tan x}[/latex]

Watch the following video to see the worked solution to Example: Using Logarithmic Differentiation.

Example: Using Logarithmic Differentiation

Find the derivative of [latex]y=\large \frac{x\sqrt{2x+1}}{e^x \sin^3 x}[/latex]

Example: Extending the Power Rule

Find the derivative of [latex]y=x^r[/latex] where [latex]r[/latex] is an arbitrary real number.

Try It

Use logarithmic differentiation to find the derivative of [latex]y=x^x[/latex].

Watch the following video to see the worked solution to the above Try It.

Try It

Find the derivative of [latex]y=(\tan x)^{\pi}[/latex].

Try It