What is (\partial)/(\partial x)(x^{x^2y}) ?

The answer to (\partial)/(\partial x)(x^{x^2y}) is yx^{x^2y}(2xln(x)+x)

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(\partial)/(\partial x)(x^{x^2y})

\frac{\partial}{\partial x} (x^{x^{2} y})

y x^{x^{2} y} (2 x ln (x) + x)

Solution steps

\frac{\partial}{\partial x} (x^{x^{2} y})

Treat

y

as a constant

Apply exponent rule:

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

x^{x^{2} y} = e^{x^{2} y l n (x)}

= \frac{\partial}{\partial x} (e^{x^{2} y l n (x)})

Apply the chain rule:

e^{x^{2} y l n (x)} \frac{\partial}{\partial x} (x^{2} y ln (x))

= e^{x^{2} y l n (x)} \frac{\partial}{\partial x} (x^{2} y ln (x))

\frac{\partial}{\partial x} (x^{2} y ln (x)) = y (2 x ln (x) + x)

= e^{x^{2} y l n (x)} y (2 x ln (x) + x)

e^{x^{2} y l n (x)} = x^{x^{2} y}

= y x^{x^{2} y} (2 x ln (x) + x)

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What is (\partial)/(\partial x)(x^{x^2y}) ?
The answer to (\partial)/(\partial x)(x^{x^2y}) is yx^{x^2y}(2xln(x)+x)