Derivative of Square Root of x to the Power x

Find ᵈʸ⁄dₓ, if y = (√x)^x.

In (√x)^x, we have varoiable x is in exponent.

To find the derivative of a term which contain variable in exponent, we have to take natural logarithm on both sides. Then, we have to use the rules of logarithm and find the derivative.

y = (√x)^x

Square root can be written as exponent ½.

Take natural logarithm on both sides.

Now, we have find the derivative on both sides with respect to x.

To find the derivative of ln(y) with respect to x, we have to use use chain rule and to find the derivative xlnx on the right side, we have to use product rule.

Multiply both sides by y.

Substitute y =(√x)^x.

Therefore,

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