Differentiate w.r.t.x:
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![y = log( \sqrt[5]{ {x}^{3} } ) \\ y = log( {x}^{ \frac{3}{5} } ) \\ diff \: with \: r \: to \: x \\ \frac{dy}{dx} = \frac{d}{dx} ( log( {x}^{ \frac{3}{5} } ) \\ = \frac{1}{ \ \sqrt[5]{ {x}^{3} } } . \frac{3}{5} . {x}^{ \frac{3}{5} - 1} \\ = \frac{3}{5 \sqrt[5]{ {x}^{3} } } . {x}^{ - \frac{2}{5} } \\ = \frac{3}{5 \sqrt[5]{ {x}^{3} } } . \frac{1}{ {x}^{ \frac{2}{5} } } \\ = \frac{3}{5( { x }^{ (\frac{3}{5} + \frac{2}{5} )} )} \\ = \frac{3}{ {x}^{ \frac{5}{5} } } \\ \frac{3}{x} ans](https://tex.z-dn.net/?f=y+%3D++log%28+%5Csqrt%5B5%5D%7B+%7Bx%7D%5E%7B3%7D+%7D+%29++%5C%5C+y+%3D++log%28+%7Bx%7D%5E%7B+%5Cfrac%7B3%7D%7B5%7D+%7D+%29++%5C%5C+diff+%5C%3A+with+%5C%3A+r+%5C%3A+to+%5C%3A+x+%5C%5C++%5Cfrac%7Bdy%7D%7Bdx%7D++%3D++%5Cfrac%7Bd%7D%7Bdx%7D+%28+log%28+%7Bx%7D%5E%7B+%5Cfrac%7B3%7D%7B5%7D+%7D+%29++%5C%5C++%3D++%5Cfrac%7B1%7D%7B+%5C+%5Csqrt%5B5%5D%7B+%7Bx%7D%5E%7B3%7D+%7D+%7D+.+%5Cfrac%7B3%7D%7B5%7D+.+%7Bx%7D%5E%7B+%5Cfrac%7B3%7D%7B5%7D+-+1%7D++%5C%5C++%3D++%5Cfrac%7B3%7D%7B5+%5Csqrt%5B5%5D%7B+%7Bx%7D%5E%7B3%7D+%7D+%7D+.+%7Bx%7D%5E%7B+-++%5Cfrac%7B2%7D%7B5%7D+%7D++%5C%5C++%3D++%5Cfrac%7B3%7D%7B5+%5Csqrt%5B5%5D%7B+%7Bx%7D%5E%7B3%7D+%7D+%7D+.++%5Cfrac%7B1%7D%7B+%7Bx%7D%5E%7B+%5Cfrac%7B2%7D%7B5%7D+%7D+%7D++%5C%5C++%3D++%5Cfrac%7B3%7D%7B5%28+%7B+x+%7D%5E%7B+%28%5Cfrac%7B3%7D%7B5%7D++%2B++%5Cfrac%7B2%7D%7B5%7D+%29%7D+%29%7D++%5C%5C++%3D++%5Cfrac%7B3%7D%7B+%7Bx%7D%5E%7B+%5Cfrac%7B5%7D%7B5%7D+%7D+%7D++%5C%5C++%5Cfrac%7B3%7D%7Bx%7D+ans "y = log( \sqrt[5]{ {x}^{3} } ) \\ y = log( {x}^{ \frac{3}{5} } ) \\ diff \: with \: r \: to \: x \\ \frac{dy}{dx} = \frac{d}{dx} ( log( {x}^{ \frac{3}{5} } ) \\ = \frac{1}{ \ \sqrt[5]{ {x}^{3} } } . \frac{3}{5} . {x}^{ \frac{3}{5} - 1} \\ = \frac{3}{5 \sqrt[5]{ {x}^{3} } } . {x}^{ - \frac{2}{5} } \\ = \frac{3}{5 \sqrt[5]{ {x}^{3} } } . \frac{1}{ {x}^{ \frac{2}{5} } } \\ = \frac{3}{5( { x }^{ (\frac{3}{5} + \frac{2}{5} )} )} \\ = \frac{3}{ {x}^{ \frac{5}{5} } } \\ \frac{3}{x} ans")
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