What is the derivative of xye^{xy} ?

The derivative of xye^{xy} is e^{xy}y+x(e^{xy}(dy)/(dx)+e^{xy}(y+x(dy)/(dx))y)

What is the first derivative of xye^{xy} ?

The first derivative of xye^{xy} is e^{xy}y+x(e^{xy}(dy)/(dx)+e^{xy}(y+x(dy)/(dx))y)

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

\frac{d x}{d x} = 1

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

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

Simplify

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

What is the derivative of xye^{xy} ?
The derivative of xye^{xy} is e^{xy}y+x(e^{xy}(dy)/(dx)+e^{xy}(y+x(dy)/(dx))y)
What is the first derivative of xye^{xy} ?
The first derivative of xye^{xy} is e^{xy}y+x(e^{xy}(dy)/(dx)+e^{xy}(y+x(dy)/(dx))y)