If y (x^x), prove that dy dx = (x^x) ^x(1+ ) - Vidyadip

Question

If $\text{y}\sin(\text{x}^\text{x}),$ prove that $\frac{\text{dy}}{\text{dx}}=\cos(\text{x}^\text{x})\times\text{x}^\text{x}(1+\log\text{x})$

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Answer

Let $\text{y}=\sin(\text{x}^\text{x})\ .....(\text{i})$
Also, Let $\text{u}=\text{x}^\text{x}\ .....(\text{ii})$
Taking log on both sides,
$\Rightarrow\log\text{u}=\log\text{x}^\text{x}$
$\Rightarrow\log\text{u}=\text{x}\log\text{x}$
Differentiating both sides with respect to x,
$\frac{1}{\text{u}}\frac{\text{du}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\text{x}\log\text{x})$
$\Rightarrow\ \frac{1}{\text{u}}\frac{\text{du}}{\text{dx}}=\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}\text{x}$
$\Rightarrow\frac{1}{\text{u}}\frac{\text{du}}{\text{dx}}=\text{x}\big(\frac{1}{\text{x}}\big)+\log\text{x}(1)$
$\Rightarrow\ \frac{1}{\text{u}}\frac{\text{du}}{\text{dx}}=1+\log\text{x}$
$\Rightarrow\frac{\text{du}}{\text{dx}}=\text{u}(1+\log\text{x})$
$\Rightarrow\frac{\text{du}}{\text{dx}}=\text{x}^\text{x}(1+\log\text{x})\ .....(\text{iii})$
[Using equation (ii)]
Now, using equation (ii) in equation (i),
$\text{y}=\sin\text{u}$
Differentiating with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\sin\text{u})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\cos\text{u}\frac{\text{du}}{\text{dx}}$
Using equation (ii) and (iii),
$\frac{\text{dy}}{\text{dx}}=\cos(\text{x}^\text{x})\times\text{x}^\text{x}(1+\log\text{x})$

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