What is (\partial)/(\partial y)(10xye^{-x^2-y^2}) ?

The answer to (\partial)/(\partial y)(10xye^{-x^2-y^2}) is 10x(e^{-x^2-y^2}-2e^{-x^2-y^2}y^2)

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(\partial)/(\partial y)(10xye^{-x^2-y^2})

\frac{\partial}{\partial y} (10 x y e^{- x^{2} - y^{2}})

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

Solution steps

\frac{\partial}{\partial y} (10 x y e^{- x^{2} - y^{2}})

Treat

x

as a constant

Take the constant out:

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

= 10 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}}

= 10 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

= 10 x (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} y) y)

Simplify

10 x (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} y) y) : 10 x (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} y^{2})

= 10 x (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} y^{2})

\frac{d}{d x} (- 3 cot (1 - x^{2}))

x \to 0 lim (3 - \frac{1}{x})

f (x) = π^{x}

\frac{d}{d x} (x x^{2} - 3 x + 2)

\int 9 x - 5 d x

What is (\partial)/(\partial y)(10xye^{-x^2-y^2}) ?
The answer to (\partial)/(\partial y)(10xye^{-x^2-y^2}) is 10x(e^{-x^2-y^2}-2e^{-x^2-y^2}y^2)