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NSW Y12 Maths - Advanced Exponential and Log Function Applications of Differentiation lnx

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Applications of Differentiation lnx Theory

Applications of the differentiation of logarithmic functions involves determining equations of tangents and stationary points on these functions.\\ \begin{multicols}{2} \textbf{Example 1}\\ %10646 Find the equation of the tangent to the curve \(y = \log_e({x^2} - 1)\) at \(x=2\)\\ \textbf{Example 1 solution}\\ $\begin{aligned} y &=\log _{e}\left(x^{2}-1\right) \\ y^{\prime} &=\frac{2 x}{x^{2}-1} \\ \text{at}\quad x &=2, y=\log _{e} 3 \\ y^{\prime} &=\frac{4}{3}\\ y-\log _{e} 3&=\frac{4}{3}(x-2)\\ 3 y-3 \log _{e} 3&=4 x-8\\ 4 x-3 y+3 \log _{e} 3-8&=0 \end{aligned}$\\ \columnbreak \textbf{Example 2}\\%10647 Find the stationary point on the curve \(y = {x^2}\log_ex\) and determine its nature\\ \textbf{Example 2 solution}\\ $\begin{aligned} y &=x^{2} \log _{e} x \\ \text{Let}\ u &=x^{2} \quad v=\log _{e} x \\ d u &=2 x \quad d u=\frac{1}{x} \\ y^{\prime} &=u d v+v d u \\ &=x^{2} \times \frac{1}{x}+2 x \log_{ e} x \\ &=x+2 x \log_{ e} x\\ y^{\prime}&=x\left(1+2 \log _{e} x\right)\\ \text{let}\quad u&=x \quad v=1+2 \log _{e} x\\ d u&=1 \quad d v=\frac{2}{x}\\ y^{\prime \prime}&=x \times \frac{2}{x}+1+2 \log _{9} x\\ &=3+2 \log_{e} x\\ \text{Let}\ y^{\prime}&=0 \quad x\left(1+2 \log _{e} x\right)=0\\ 1+2 \log_{e}x&=0 \quad x \neq 0\\\ 2 \log _{e} x &=-1 \\ \log _{e} x &=-\frac{1}{2} \\ x &=e^{-1 / 2}\\ y&=e^{-1} \log _{e} e^{-\frac{1}{2}}={-\frac{1}{2e}}\\ \text{At}\ x&=e^{-\frac{1}{2}}\\ y^{\prime \prime} &=3+2 \log_{e} e^{-\frac{1}{2}} \\ &=2 \quad \therefore\ \text{min at}\ \left(e^{-\frac{1}{2}},-\frac{1}{2 e}\right) \end{aligned}$\\ \end{multicols}

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  • Applications of Differentiation lnx - Video - Equation of normal line | Derivative applications | Differential Calculus

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