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posted by  Kels1001 on 11/3/2009 12:49:18 PM  |  status: Closed  |  Earned Karma: 58

Similar question: Lifesaver! Composition derivative. Please help!!

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Calculus Calculus Single and Multivariable (5th) by Hughes-Hallett, Gleason, McCallum N/A N/A 11/4/2009 at 12:00:00 PM
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posted by anonymousxyz on 11/3/2009 4:36:02 PM  |  status: Live
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\begin{align*} & j(x) = (g^{-1} \circ f)(x) = g^{-1}(\;f(x)\;) \\\\& j'(x) = {g^{-1}} ' (\;f(x)\;)f'(x) \end{align*}


\begin{align*} j'( 5 ) = {g^{-1}} ' (\;f(5)\;)f'(5) = {g^{-1}} ' (3)({-2}) \end{align*}


\text{Find } {g^{-1}} '( 3 ) \text{:}


\begin{align*} & g(\;g^{-1}(x)\;)= x &\Rightarrow\\\\& \frac{\text{d}}{\text{d}x}\;g(\;g^{-1}(x)\;) = \frac{\text{d}}{\text{d}x}\;x &\Rightarrow\\\\& g'(\;g^{-1}(x)\;){g^{-1}}'(x) = 1 &\Rightarrow\\\\& {g^{-1}}'(x) = \frac{1}{g'(\;g^{-1}(x)\;)}&\Rightarrow\\\\& {g^{-1}}'(3) = \frac{1}{g'(\;g^{-1}(3)\;)} \end{align*}


\text{Find } g^{-1}(3) \text{:}


\begin{align*} & g(\;g^{-1}(x)\;) = x \qquad \Rightarrow \qquad g(\;g^{-1}(3)\;) = 3 \end{align*}


\text{ Find the value of } x \text{ for which } g( x ) = 3\text{:}


g( 4 ) = 3 \qquad \Rightarrow \qquad g^{-1}(3) = 4


{g^{-1}}'(3) = \frac{1}{g'(\;g^{-1}(3)\;)} = \frac{1}{g'(4)} = \frac{1}{5}


\begin{align*} j'( 5 ) = {g^{-1}} ' (3)(-2) = \bigg(\frac{1}{5}\bigg)\bigg({-2}\bigg) = -\frac{2}{5} \end{align*}
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