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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

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