logarithmic derivative

Given a function $f$, the quantity $f^{\prime }/f$ is known as thelogarithmic derivative of $f$. This name comesfrom the observation that, on account of the chainrule,

$${(\log f)}^{\prime }=f^{\prime }{\log }^{\prime }(f)=f^{\prime }/f.$$

The logarithmic derivative has several basic propertieswhich make it useful in various contexts.

The logarithmic derivative of the product offunctions is the sum of their logarithmicderivatives. This follows from the product rule:

$$\frac{{(fg)}^{\prime }}{fg}=\frac{f{g}^{\prime }+f^{\prime }g}{fg}=\frac{f^{\prime }}{f}+\frac{g^{\prime }}{g}$$

The logarithmic derivative of the quotient offunctions is the difference of their logarithmicderivatives. This follows from the quotient rule:

$$\frac{{(f/g)}^{\prime }}{f/g}=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}\frac{g}{f}=\frac{f^{\prime }}{f}-\frac{g^{\prime }}{g}$$

The logarithmic derivative of the $p$-th powerof a function is $p$ times the logarithmicderivative of the function. This followsfrom the power rule:

$$\frac{{(f^p)}^{\prime }}{f^p}=\frac{p{f}^{p-1}{f}^{\prime }}{f^p}=p\frac{f^{\prime }}{f}$$

The logarithmic derivative of the exponentialof a function equals the derivative of afunction. This follows from the chain rule:

$$\frac{{\left(e^f\right)}^{\prime }}{e^f}=\frac{e^f{f}^{\prime }}{e^f}=f^{\prime }$$

Using these identities, it is rather easy tocompute the logarithmic derivatives of expressionswhich are presented in factored form. For instance,suppose we want to compute the logarithmic derivativeof

$$e^{x^2}\frac{{(x-2)}^3(x-3)}{x-1}.$$

Using our identities, we find that its logarithicderivative is

$$2x+\frac{3}{x-2}+\frac{1}{x-3}-\frac{1}{x-1}.$$