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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

\int \frac{x ^{2} + 3}{x x ^{2} - 9} d x

\frac{\partial}{\partial x} (cos^{6} (x^{8} y^{6}))

y = 2 x - x^{2}, y = 2 x - 4

x \to 9 lim \frac{( x ^{\frac{1}{2}} - 3 ) ( x + 1 )}{x ^{2} - 8 x - 9}

y = (3 x^{2} - 2 x - 4) 5^{x}

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