📂Functions

Derivatives of Logarithmic Functions

Formulas

The derivative of a logarithmic function with base $e$ is as follows.

$$\begin{equation} \dfrac{d \log x}{dx}=\dfrac{1}{x} \end{equation}$$

The derivative of a composite logarithmic function is as follows.

$$\begin{equation} \dfrac{d \left( \log f(x) \right)}{dx} = \dfrac{f^{\prime}(x)}{f(x)} \end{equation}$$

Explanation

Especially, $(2)$ is used as a useful substitution trick.

Derivation

$(1)$

By the definition of logarithmic functions, the following equation holds.

$$x = e^{\log x}$$

Differentiating both sides results in the following, by the derivative of the exponential function and chain rule.

$$\begin{align*} 1 &= \dfrac{ d \left( e^{\log x} \right) }{dx} \\ &= \dfrac{ d \left( e^{\log x} \right) }{d \log x} \dfrac{d \log x}{dx} \\ &= e^{\log x} \dfrac{d \log x}{dx} \\ &= x \dfrac{d \log x}{dx} \end{align*}$$

$$\implies \dfrac{d \log x}{dx} = \dfrac{1}{x}$$

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$(2)$

By the definition of logarithmic functions, the following equation holds.

$$f(x) = e^{\log f(x)}$$

The remaining process is the same as above.

$$\begin{align*} f^{\prime} &= \dfrac{ d e^{\log f(x)}}{dx} \\ &= \dfrac{ d e^{\log f(x)}}{d \log f(x)} \dfrac{d \log f(x)}{dx} \\ &= e^{\log f(x)} \dfrac{d \log f(x)}{dx} \\ &= f(x) \dfrac{d \log f(x)}{dx} \end{align*}$$

$$\implies \dfrac{d \log f(x)}{dx} = \dfrac{f^{\prime}(x)}{f(x)}$$

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