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

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

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

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

y (e^{x y} + e^{x y} x y)

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = e^{x y}

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

\frac{\partial}{\partial x} (x) = 1

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

= y (1 \cdot e^{x y} + e^{x y} y x)

Simplify

= y (e^{x y} + e^{x y} x y)

\frac{d}{d x} (3 x^{2} - e^{x})

\int 4^{t} sin (4^{t}) d t

\frac{d}{d x} (y - x + 1 + x - y + 1)

f (x) = ln (x + 9)

- 4 ln (+ \frac{1}{x} - 4)

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