What is (\partial)/(\partial x)(5xye^{-x^2y}) ?

The answer to (\partial)/(\partial x)(5xye^{-x^2y}) is 5y(e^{-x^2y}-2e^{-x^2y}x^2y)

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(\partial)/(\partial x)(5xye^{-x^2y})

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

= 5 y \frac{\partial}{\partial 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}

= 5 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}) = - 2 y e^{- y x^{2}} x

= 5 y (1 \cdot e^{- x^{2} y} + (- 2 y e^{- y x^{2}} x) x)

Simplify

5 y (1 \cdot e^{- x^{2} y} + (- 2 y e^{- y x^{2}} x) x) : 5 y (e^{- x^{2} y} - 2 e^{- x^{2} y} x^{2} y)

= 5 y (e^{- x^{2} y} - 2 e^{- x^{2} y} x^{2} y)

What is (\partial)/(\partial x)(5xye^{-x^2y}) ?
The answer to (\partial)/(\partial x)(5xye^{-x^2y}) is 5y(e^{-x^2y}-2e^{-x^2y}x^2y)