What is (\partial)/(\partial y)(5x^{4y}) ?

The answer to (\partial)/(\partial y)(5x^{4y}) is 20x^{4y}ln(x)

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(\partial)/(\partial y)(5x^{4y})

\frac{\partial}{\partial y} (5 x^{4 y})

20 x^{4 y} ln (x)

Solution steps

\frac{\partial}{\partial y} (5 x^{4 y})

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 \frac{\partial}{\partial y} (x^{4 y})

Apply exponent rule:

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

x^{4 y} = e^{4 y l n (x)}

= 5 \frac{\partial}{\partial y} (e^{4 y l n (x)})

Apply the chain rule:

e^{4 y l n (x)} \frac{\partial}{\partial y} (4 y ln (x))

= e^{4 y l n (x)} \frac{\partial}{\partial y} (4 y ln (x))

\frac{\partial}{\partial y} (4 y ln (x)) = 4 ln (x)

= 5 e^{4 y l n (x)} \cdot 4 ln (x)

Simplify

5 e^{4 y l n (x)} \cdot 4 ln (x) : 20 x^{4 y} ln (x)

= 20 x^{4 y} ln (x)

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What is (\partial)/(\partial y)(5x^{4y}) ?
The answer to (\partial)/(\partial y)(5x^{4y}) is 20x^{4y}ln(x)