What is the d/(dy)(ye^{xy}) ?

The d/(dy)(ye^{xy}) is e^{yx}+e^{yx}yx

What is the first d/(dy)(ye^{xy}) ?

The first d/(dy)(ye^{xy}) is e^{yx}+e^{yx}yx

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

e^{y x} + e^{y x} y x

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

\frac{d y}{d y} = 1

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

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

Simplify

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

What is the d/(dy)(ye^{xy}) ?
The d/(dy)(ye^{xy}) is e^{yx}+e^{yx}yx
What is the first d/(dy)(ye^{xy}) ?
The first d/(dy)(ye^{xy}) is e^{yx}+e^{yx}yx