Tag Archives: Logarithmic Functions

Derivative of Logarithmic Functions

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 $y=\log_e{(x^2+1)}$. \( \begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}\log_e{(x^2+1)} \\ &= \dfrac{1}{x^2+1} \times \dfrac{d}{dx}(x^2+1) \\ &= \dfrac{1}{x^2+1} \times 2x \\ &= \dfrac{2x}{x^2+1} \\ \end{align} \) Example 2 Find $\displaystyle \dfrac{dy}{dx}$ if $y=x^2\log_e{(2x-1)}$. \( \begin{align} \displaystyle \require{color} \dfrac{dy}{dx} &= \dfrac{d}{dx}x^2 \times \log_e{(2x-1)} + […]

Logarithmic Differentiation

Logarithmic Differentiation

Basic Rule of Logarithmic Differentiation $$ \displaystyle \dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \\ \dfrac{d}{dx}\log_e{f(x)} = \dfrac{f'(x)}{f(x)} $$ Practice Questions Question 1 Differentiate \( y = \log_{e}(3x) \). \( \begin{aligned} \displaystyle \dfrac{d}{dx}\log_{e}(3x) &= \dfrac{(3x)’}{3x} \\ &= \dfrac{3}{3x} \\ &= \dfrac{1}{x} \end{aligned} \) Question 2 Differentiate \( y = \log_{e}(2x-1) \). \( \begin{aligned} \displaystyle \dfrac{d}{dx}\log_{e}(2x-1) &= \dfrac{(2x-1)’}{2x-1} \\ &= […]