What is the derivative of sqrt(x)e^{xy} ?

The derivative of sqrt(x)e^{xy} is (e^{xy})/(2sqrt(x))+e^{xy}sqrt(x)(y+x(dy)/(dx))

What is the first derivative of sqrt(x)e^{xy} ?

The first derivative of sqrt(x)e^{xy} is (e^{xy})/(2sqrt(x))+e^{xy}sqrt(x)(y+x(dy)/(dx))

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

\frac{e ^{x y}}{2 x} + e^{x y} x (y + x \frac{d y}{d x})

Solution steps

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

Apply the Product Rule:

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

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

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

\frac{d}{d x} (x) = \frac{1}{2 x}

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

= \frac{1}{2 x} e^{x y} + e^{x y} (y + x \frac{d y}{d x}) x

\frac{1}{2 x} e^{x y} = \frac{e ^{x y}}{2 x}

= \frac{e ^{x y}}{2 x} + e^{x y} x (y + x \frac{d y}{d x})

What is the derivative of sqrt(x)e^{xy} ?
The derivative of sqrt(x)e^{xy} is (e^{xy})/(2sqrt(x))+e^{xy}sqrt(x)(y+x(dy)/(dx))
What is the first derivative of sqrt(x)e^{xy} ?
The first derivative of sqrt(x)e^{xy} is (e^{xy})/(2sqrt(x))+e^{xy}sqrt(x)(y+x(dy)/(dx))