Derivative of Logarithmic Functions

$$\dfrac{d}{dx}\log_e{x} = \dfrac{1}{x}$$ $$\dfrac{d}{dx}\log_e{f(x)} = \dfrac{1}{f(x)} \times f'(x)$$

Example 1

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\log_e{(x^2+1)}$.

Example 2

Find $\displaystyle \dfrac{dy}{dx}$ if $y=x^2\log_e{(2x-1)}$.
The laws of logarithmic can help to find the derivative of logarithmic Functions more easily.
$$ \begin{align} \log_e{(ab)} &= \log_e{a} + \log_e{b} \\ \log_e{\dfrac{a}{b}} &= \log_e{a} - \log_e{b} \\ \log_ea^n &= n\log_e{a} \\ \end{align} $$

Example 3

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\log_e{(x^2+1)(x^3-1)}$.

Example 4

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\log_e{\sqrt{x^2+2}}$.

Example 5

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\log_e{\dfrac{x^2}{(x+3)(x-5)}}$.

Example 6

Compare the derivatives of $\log_e{x^3}$ and $(\log_e{x})^3$.

Extension Examples

These Extension Examples require to have some prerequisite skills including;
\( \begin{align} \displaystyle \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \dfrac{d}{dx}\cos{x} &= -\sin{x} \\ \dfrac{d}{dx}e^x &= e^x \\ \end{align} \)

Example 7

Find $\displaystyle \dfrac{dy}{dx}$ of $e^x\log_e{x}$, known that $\dfrac{d}{dx}e^x = e^x$.

Example 8

Find $\displaystyle \dfrac{dy}{dx}$ of $\displaystyle y=\log_e{\dfrac{\sin{x}}{\cos{x}}}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$ and $\dfrac{d}{dx}\cos{x} = -\sin{x}$.

Example 9

Find $\displaystyle \dfrac{dy}{dx}$ of $\displaystyle y=\log_e{\sin{x^2}}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$.