Derivatives of Logs

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Definition of derivative

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Derivatives of Logs

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Logarithmic Differentiation

Derivatives in Science

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Logarithmic Differentiation

Now that we know the derivative of a log, we can combine it with the chain rule:

$\frac{d}{dx}\Big( \ln(y)\Big)= \frac{1}{y} \frac{dy}{dx},$ or equivalently

$\frac{dy}{dx} = y \frac{d }{dx}\Big(\ln(y)\Big).$ Sometimes it is (much!) easier to take the derivative of

$\ln(y)$ than of

$y$ . In those cases, we can use the last equation to get

$dy/dx$ . This is called logarithmic differentiation.

Example: Find the derivative of $y=x^x$ .

Solution: Take the log of both sides to get $\ln(y) = x \ln(x)$ . By the product rule, the derivative of $x \ln(x)$ is $\ln(x) + 1$ . So $\frac{dy}{dx} = y \frac{d}{dx}\Big(\ln(y)\Big) = x^x (1+\ln(x)).$