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• Find The Derivative of x ¹/x

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❍ When dealing with a function raised to the power of function, lagorithmic differentiation becomes necessary

Let,

$\begin{gathered} \bold{y = {x}^{ \frac{1}{x} }} \end{gathered}$

Then,

$\sf{ \qquad : \implies \quad \: y = ln( {x}^{ \frac{1}{x} } ) }$

$\bold{ \underline{ \: {Recalling \: that \: ln( {x}^{a} ) = a \: ln \: x : }}}$

$\sf{ \qquad : \implies \quad \: ln \: y \: = \: \frac{1}{x} ln \: x}$

$\sf{ \qquad : \implies \quad \: ln \: y \: = \: \large \frac{ ln \: x }{x} }$

❍ Now, Differentiate both the side with respect to x, meaning that the left side will be implicitly differentiated :

$\begin{gathered} \sf{ \frac{1}{y} \: . \: \frac{dy}{dx} = \frac{1 - ln \: x }{ {x}^{2} }} \end{gathered}$

$\bold{ \underline{ \: {solve \: for \: \frac{dy}{dx} : }}}$

$\begin{gathered} \: \sf \: \frac{dy}{dx} = y \: ({ \frac{1 - ln \: x }{ {x}^{2} })} \end{gathered}$

$\bold{ \underline{write \: everything \: in \: terms \: of \: x : }}$

$\begin{gathered} \sf \: \frac{dy}{dx} = {x}^{ \frac{1}{x} } \: \:( \frac{1 - ln \: x }{ {x}^{2} } ) \end{gathered}$