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

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Differentiation of Exponential Functions Theory

This subtopic is a revision of the subtopic exponential functions in the year 11 course.\\ The derivative of \(y=e^{f(x)}\) is \(y^{\prime}=f^{\prime}(x) e^{f(x)}\)\\ \begin{multicols}{2} \textbf{Example 1}\\ %10622 Differentiate \(y = {({e^x} + {e^{ - x}})^2}\)\\ \textbf{Example 1 solution}\\ $\begin{aligned} y&=\left(e^{x}+e^{-x}\right)^{2} \\ y^{\prime} &=2\left(e^{x}+e^{-x}\right)\left(e^{x}-e^{-x}\right) \\ &=2\left(e^{2 x}-e^{-2 x}\right) \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ %124266 Differentiate \(y = \dfrac{{{e^x}}}{{{e^{ - x}} + 2}}\)\\ \textbf{Example 2 solution}\\ \(\begin{aligned} y &=\frac{e^{x}}{e^{-x}+2} \end{aligned}\)\\ \(\begin{aligned} \text {Let } u &=e^{x} & v&=e^{-x}+2 \\ du &=e^{x} & dv&=-e^{-x} \end{aligned}\)\\ \(\begin{aligned} \frac{d y}{d x} &=\frac{v d u-u d v}{v^{2}} \\ &=\frac{e^{x}\left(e^{-x}+2\right)--e^{-x} \times e^{x}}{\left(e^{-x}+2\right)^{2}} \\ &=\frac{e^{0}+2 e^{x}+e^{0}}{\left(e^{-x}+2\right)^{2}} \\ &=\frac{2\left(1+e^{x}\right)}{\left(e^{-x}+2\right)^{2}} \end{aligned}\) \end{multicols}

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Videos relating to Differentiation of Exponential Functions.

  • Differentiation of Exponential Functions - Video - Differentiation of exponential functions

  • Differentiation of Exponential Functions - Video - Differentiating Exponential Functions | HSC Advanced Mathematics

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