Question

.Describe the process called logarithmic differentiation. What types of differentiation problems is logarithmic differentiation useful for?

Short answer

In calculus, logarithmic differentiation is a method of differentiation that employs the logarithmic derivative of a function of $f.$

The logarithmic function is useful when applies to a function raised to the power of variables or functions.

Step-by-step solution

Step 1. Given information

We need to describe the process called logarithmic differentiation and also we need to write the types of differentiation problems is logarithmic differentiation useful.

Step 2. Explanation

In calculus, logarithmic differentiation is a method of differentiation that employs the logarithmic derivative of a function of $f.$

For example,

$f\left(x\right)=x^x.$

Take log both sides.

role="math" $\log f\left(x\right)=\log x^x.$

$\log f\left(x\right)=x\log x.$

Differentiating both sides with respect to $x.$

$\frac{1}{f\left(x\right)}\times f^{\text{'}}\left(x\right)=x\times \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.+\log x.$

$f^{\text{'}}\left(x\right)=\left(1+\log x\right)f\left(x\right).$

$f^{\text{'}}\left(x\right)=x^x\left(1+\log x\right).$

The logarithmic function is useful when applies to a function raised to the power of variables or functions like,$f\left(x\right)=e^{\left(e^x\right)},orf\left(x\right)={\left(e^x\right)}^e.$