Differentiate the following function with respect to `x:` `x^(cos^-1x)`

Explanation

Let `{y}=x^{cos ^{-1} x} ldots(i)`

Taking log both sides

&`log y=log x^{cos ^{-1} x}`

&`Rightarrow log y=cos ^{-1} x log x`

`frac{1}{y} frac{d y}{d x}=cos ^{-1} x frac{d}{d x}(log x)+log x frac{d}{d x} cos ^{-1} x`

`Rightarrow frac{1}{y} frac{d y}{d x}=cos ^{-1} x(frac{1}{x})+log x(frac{-1}{sqrt{1-x^{2}}})`

`Rightarrow frac{1}{y} frac{d y}{d x}=frac{cos ^{-1} x}{x}-frac{log x}{sqrt{1-x^{2}}}`

`Rightarrow frac{d y}{d x}=y[frac{cos ^{-1} x}{x}-frac{log x}{sqrt{1-x^{2}}}]`

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