What is the derivative of xe^{x^2y} ?

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

What is the first derivative of xe^{x^2y} ?

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

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

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

Solution steps

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

Apply the Product Rule:

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

f = x, g = e^{x^{2} y}

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

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

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

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

Simplify

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

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