What is (\partial)/(\partial y)(x^{9y}) ?

The answer to (\partial)/(\partial y)(x^{9y}) is 9x^{9y}ln(x)

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(\partial)/(\partial y)(x^{9y})

\frac{\partial}{\partial y} (x^{9 y})

9 x^{9 y} ln (x)

Solution steps

\frac{\partial}{\partial y} (x^{9 y})

Treat

x

as a constant

Apply exponent rule:

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

x^{9 y} = e^{9 y l n (x)}

= \frac{\partial}{\partial y} (e^{9 y l n (x)})

Apply the chain rule:

e^{9 y l n (x)} \frac{\partial}{\partial y} (9 y ln (x))

= e^{9 y l n (x)} \frac{\partial}{\partial y} (9 y ln (x))

\frac{\partial}{\partial y} (9 y ln (x)) = 9 ln (x)

= e^{9 y l n (x)} \cdot 9 ln (x)

e^{9 y l n (x)} = x^{9 y}

= 9 x^{9 y} ln (x)

\int \frac{x ^{2}}{x ^{3} - x ^{2} + 4 x - 4} d x

\int x arctan (3 x) d x

x \to 0 - lim (\frac{∣ x - 7 ∣}{x - 7})

y^{^{''}} + 8 y^{^{'}} + 16 y = 0, y (0) = 3, y^{^{'}} (0) = 8

8 3 x^{2} + 9

What is (\partial)/(\partial y)(x^{9y}) ?
The answer to (\partial)/(\partial y)(x^{9y}) is 9x^{9y}ln(x)