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NSW Y12 Maths - Advanced Exponential and Log Function Differentiation of Log Functions

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Differentiation of Log Functions Theory

\(y=\log _e f(x)\) then \(\dfrac{d y}{d x}=\dfrac{f^{\prime}(x)}{f(x)}\)\\ Knowing the logarithm laws will also help in the differentiation of logarithmic functions\\ \(\log a+\log b=\log a b\)\\ \(\log \dfrac{a}{b}=\log a-\log b \)\\ \(\log a^n=n \log a\)\\ \textbf{Different bases:} \\ \(y=\log _a f(x)\) then \(\dfrac{d y}{d x}=\dfrac{1}{\ln a} \times \dfrac{f^{\prime}(x)}{f(x)}\)\\ \begin{multicols}{2} \textbf{Example 1}\\ %24741 Differentiate \(\ln \sqrt {x + 1} \)\\ \textbf{Example 1 solution}\\ $\begin{aligned} \dfrac{d}{dx}( \ln \sqrt{x+1})&=\dfrac{d}{dx}(\frac{1}{2} \ln (x+1))\\ &=\frac{1}{2} \times \dfrac{1}{x+1}\\ &=\dfrac{1}{2(x+1)} \end{aligned}$\\ \textbf{Example 2}\\ %24743 Differentiate \(\ln x{(x + 1)^3}\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} \text { let } y &=\ln x(x+1)^{3} \\ &=\ln x+3 \ln (x+1) \\ y^{\prime}=& \frac{1}{x}+\frac{3}{x+1} \\ =& \frac{x+1+3 x}{x(x+1)} \\ =& \frac{4 x+1}{x(x+1)} \end{aligned}$\\ \columnbreak \textbf{Example 3}\\ %24743 Differentiate \(y=\log _2(3 x+1)\) \\ \textbf{Example 3 solution}\\ $\begin{aligned} \frac{d y}{d x} & =\frac{1}{\ln 2} \times \frac{3}{(3 x+1)} \\ & =\frac{3}{\ln 2(3 x+1)} \end{aligned}$\\ \end{multicols}

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