What is (\partial)/(\partial y)(x^{3y}) ?

The answer to (\partial)/(\partial y)(x^{3y}) is 3x^{3y}ln(x)

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(\partial)/(\partial y)(x^{3y})

\frac{\partial}{\partial y} (x^{3 y})

3 x^{3 y} ln (x)

Solution steps

\frac{\partial}{\partial y} (x^{3 y})

Treat

x

as a constant

Apply exponent rule:

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

x^{3 y} = e^{3 y l n (x)}

= \frac{\partial}{\partial y} (e^{3 y l n (x)})

Apply the chain rule:

e^{3 y l n (x)} \frac{\partial}{\partial y} (3 y ln (x))

= e^{3 y l n (x)} \frac{\partial}{\partial y} (3 y ln (x))

\frac{\partial}{\partial y} (3 y ln (x)) = 3 ln (x)

= e^{3 y l n (x)} \cdot 3 ln (x)

e^{3 y l n (x)} = x^{3 y}

= 3 x^{3 y} ln (x)

x \to 0 lim (a^{\frac{1}{x}})

\int x e^{(x + 2)} d x

y^{^{''}} + 7 y^{^{'}} + 10 y = 0

\int - sinh (x) d x

\frac{d}{d x} (7 x + s + 9)

What is (\partial)/(\partial y)(x^{3y}) ?
The answer to (\partial)/(\partial y)(x^{3y}) is 3x^{3y}ln(x)