What is (\partial)/(\partial y)(x^{5y}) ?

The answer to (\partial)/(\partial y)(x^{5y}) is 5x^{5y}ln(x)

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(\partial)/(\partial y)(x^{5y})

\frac{\partial}{\partial y} (x^{5 y})

5 x^{5 y} ln (x)

Solution steps

\frac{\partial}{\partial y} (x^{5 y})

Treat

x

as a constant

Apply exponent rule:

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

x^{5 y} = e^{5 y l n (x)}

= \frac{\partial}{\partial y} (e^{5 y l n (x)})

Apply the chain rule:

e^{5 y l n (x)} \frac{\partial}{\partial y} (5 y ln (x))

= e^{5 y l n (x)} \frac{\partial}{\partial y} (5 y ln (x))

\frac{\partial}{\partial y} (5 y ln (x)) = 5 ln (x)

= e^{5 y l n (x)} \cdot 5 ln (x)

e^{5 y l n (x)} = x^{5 y}

= 5 x^{5 y} ln (x)

What is (\partial)/(\partial y)(x^{5y}) ?
The answer to (\partial)/(\partial y)(x^{5y}) is 5x^{5y}ln(x)