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

Question:

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}.

Answer and Explanation: 1