What is (\partial)/(\partial y)(x^{y-1}) ?

The answer to (\partial)/(\partial y)(x^{y-1}) is x^{y-1}ln(x)

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(\partial)/(\partial y)(x^{y-1})

\frac{\partial}{\partial y} (x^{y - 1})

x^{y - 1} ln (x)

Solution steps

\frac{\partial}{\partial y} (x^{y - 1})

Treat

x

as a constant

Apply exponent rule:

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

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

= \frac{\partial}{\partial y} (e^{(y - 1) l n (x)})

Apply the chain rule:

e^{(y - 1) l n (x)} \frac{\partial}{\partial y} ((y - 1) ln (x))

= e^{(y - 1) l n (x)} \frac{\partial}{\partial y} ((y - 1) ln (x))

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

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

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

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

What is (\partial)/(\partial y)(x^{y-1}) ?
The answer to (\partial)/(\partial y)(x^{y-1}) is x^{y-1}ln(x)