Derivative of x to the Power of Sinx

Find ᵈʸ⁄dₓ, if y = x^sinx.

In x^sinx, 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^sinx

Take natural logarithm on both sides.

ln(y) = ln(x^sinx)

ln(y) = sinx ⋅ ln(x)

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.

To find the derivative sinx⋅ln(x) on the right side, we have to use product rule.

Multiply both sides by y.

Substitute y = x^sinx.

Therefore,

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