$$ \large \dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \\\dfrac{d}{dx}\log_e{f(x)} = \dfrac{1}{f(x)} \times f'(x)$$ 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{AMSsymbols} \require{color}\dfrac{dy}{dx} &= \dfrac{d}{dx}x^2 \times \log_e{(2x-1)} + x^2 \times \dfrac{d}{dx}\log_e{(2x-1)} &\color{red} […]
