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

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

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(\partial)/(\partial x)((xy)^{xy})

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

x^{x y} y^{x y + 1} (ln (x y) + 1)

Solution steps

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

Treat

y

as a constant

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

e^{x y l n (x y)} y (ln (y x) + 1) : x^{x y} y^{x y + 1} (ln (x y) + 1)

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

\int 3 - tan (\frac{x}{6}) d x

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

\int \frac{8 x ^{2} + x + 8}{( x ^{2} + 1 ) ^{2}} d x

f (x) = 2 e^{x}, at x = 2

\int 9 (3 x + 2)^{2} d x

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