Chapter 3 - Determining Change: Derivatives - 3.4 Activities - Page 224: 26

inside: $g(x)=1+9.8 e^{-1.2x}$ outside: $f(g)=37.5(g(x) )^{-1}$ derivative: $f^{\prime}(x)=441e^{-1.2x}(1+9.8 e^{-1.2 x} )^{-2}$

Work Step by Step

Given$$f(x)=\frac{37.5}{1+9.8 e^{-1.2 x}}$$ Rewriting $f(x)$ as $$f(x)= 37.5(1+9.8 e^{-1.2x})^{-1}$$ Use the chain rule to take the derivative $$\frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x)$$ Here $g(x)=1+9.8 e^{-1.2x}$ and $f(g)= 37.5(g(x) )^{-1},$ then \begin{align*} f^{\prime}(x) &=\left(37.5(g(x) )^{-1}\right)^{\prime} \\ &=-37.5(g(x) )^{-2}g^{\prime}(x) \\ &=-37.5(1+9.8 e^{-1.2 x} )^{-2}((-1.2)9.8e^{-1.2 x}) \\ &=441e^{-1.2x}(1+9.8 e^{-1.2 x} )^{-2} \end{align*}

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