If {eq}f(x) = (\sqrt x)^x {/eq} find {eq}f'(x) {/eq}


If {eq}f(x) = (\sqrt x)^x {/eq} find {eq}f'(x) {/eq}

Logarithmic Differentiation:

To differentiate a complicated function of the form {eq}\displaystyle y=f(x)^{g(x)} {/eq}, we first take natural logarithm of both sides of the equation {eq}\displaystyle y=f(x)^{g(x)} {/eq} to get {eq}\displaystyle \ln y=g(x) \ln f(x) {/eq}.

Then we differentiate both sides of the last equation to get {eq}\displaystyle \frac {1}{y} \frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\mathrm{d} }{\mathrm{d} x} (g(x) \ln f(x)) \Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x} = y \frac{\mathrm{d} }{\mathrm{d} x} (g(x) \ln f(x)) = f(x)^{g(x)}\frac{\mathrm{d} }{\mathrm{d} x} (g(x) \ln f(x)) {/eq}.

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Answer: {eq}(\sqrt x)^x(0.5+\ln (\sqrt x)) {/eq}


Let {eq}y=f(x)=(\sqrt x)^x {/eq}.

Then {eq}\ln y = x\ln (\sqrt x)=0.5x\ln...

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Differentiation Strategy: Definition & Examples


Chapter 7 / Lesson 15

What is a differentiation strategy? Learn about focused differentiation strategy, broad differentiation strategy, and other differentiation strategy examples.

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