What is (\partial)/(\partial x)(x^{xy}) ?

The answer to (\partial)/(\partial x)(x^{xy}) is yx^{xy}(ln(x)+1)

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(\partial)/(\partial x)(x^{xy})

\frac{\partial}{\partial x} (x^{x y})

y x^{x y} (ln (x) + 1)

Solution steps

\frac{\partial}{\partial x} (x^{x y})

Treat

y

as a constant

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{x y} = e^{x y l n (x)}

= \frac{\partial}{\partial x} (e^{x y l n (x)})

Apply the chain rule:

e^{x y l n (x)} \frac{\partial}{\partial x} (x y ln (x))

= e^{x y l n (x)} \frac{\partial}{\partial x} (x y ln (x))

\frac{\partial}{\partial x} (x y ln (x)) = y (ln (x) + 1)

= e^{x y l n (x)} y (ln (x) + 1)

e^{x y l n (x)} = x^{x y}

= y x^{x y} (ln (x) + 1)

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What is (\partial)/(\partial x)(x^{xy}) ?
The answer to (\partial)/(\partial x)(x^{xy}) is yx^{xy}(ln(x)+1)