What is (\partial)/(\partial x)(xe^{(x^2-y)}) ?

The answer to (\partial)/(\partial x)(xe^{(x^2-y)}) is e^{x^2-y}+2e^{x^2-y}x^2

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(\partial)/(\partial x)(xe^{(x^2-y)})

\frac{\partial}{\partial x} (x e^{(x^{2} - y)})

e^{x^{2} - y} + 2 e^{x^{2} - y} x^{2}

Solution steps

\frac{\partial}{\partial x} (x e^{(x^{2} - y)})

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

\frac{\partial}{\partial x} (e^{x^{2} - y}) = e^{x^{2} - y} \cdot 2 x

= 1 \cdot e^{x^{2} - y} + e^{x^{2} - y} \cdot 2 xx

Simplify

1 \cdot e^{x^{2} - y} + e^{x^{2} - y} \cdot 2 xx : e^{x^{2} - y} + 2 e^{x^{2} - y} x^{2}

= e^{x^{2} - y} + 2 e^{x^{2} - y} x^{2}

\int (x + 2) (3 x^{2} + 12 x) d x

\frac{\partial}{\partial y} (x y^{2} + y e^{x^{2}} + 5)

y = ln (x + 5)

x \to 2 lim (x^{2} + 6)

\frac{( 4 x ^{2} + 4 x + 6 )}{x}

What is (\partial)/(\partial x)(xe^{(x^2-y)}) ?
The answer to (\partial)/(\partial x)(xe^{(x^2-y)}) is e^{x^2-y}+2e^{x^2-y}x^2