What is (\partial)/(\partial y)(3xye^{-x^2-y^2}) ?

The answer to (\partial)/(\partial y)(3xye^{-x^2-y^2}) is 3x(e^{-x^2-y^2}-2e^{-x^2-y^2}y^2)

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(\partial)/(\partial y)(3xye^{-x^2-y^2})

\frac{\partial}{\partial y} (3 x y e^{- x^{2} - y^{2}})

3 x (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} y^{2})

Solution steps

\frac{\partial}{\partial y} (3 x y e^{- x^{2} - y^{2}})

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 x \frac{\partial}{\partial y} (y e^{- x^{2} - y^{2}})

Apply the Product Rule:

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

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

= 3 x (\frac{\partial}{\partial y} (y) e^{- x^{2} - y^{2}} + \frac{\partial}{\partial y} (e^{- x^{2} - y^{2}}) y)

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

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

= 3 x (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} y) y)

Simplify

3 x (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} y) y) : 3 x (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} y^{2})

= 3 x (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} y^{2})

\int sin (5 x) sin (2 x) d x

\frac{\partial}{\partial x} (7 x y^{2})

x \to - 3 lim (- 4 x + 6)

\int cot^{4} (3 x) d x

\frac{x ^{3}}{10 ( x - 2 )}

What is (\partial)/(\partial y)(3xye^{-x^2-y^2}) ?
The answer to (\partial)/(\partial y)(3xye^{-x^2-y^2}) is 3x(e^{-x^2-y^2}-2e^{-x^2-y^2}y^2)