What is (\partial)/(\partial x)(15xye^{-x^2-y^2}) ?

The answer to (\partial)/(\partial x)(15xye^{-x^2-y^2}) is 15y(e^{-x^2-y^2}-2e^{-x^2-y^2}x^2)

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(\partial)/(\partial x)(15xye^{-x^2-y^2})

\frac{\partial}{\partial x} (15 x y e^{- x^{2} - y^{2}})

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

Solution steps

\frac{\partial}{\partial x} (15 x y e^{- x^{2} - y^{2}})

Treat

y

as a constant

Take the constant out:

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

= 15 y \frac{\partial}{\partial x} (x e^{- x^{2} - y^{2}})

Apply the Product Rule:

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

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

= 15 y (\frac{\partial}{\partial x} (x) e^{- x^{2} - y^{2}} + \frac{\partial}{\partial x} (e^{- x^{2} - y^{2}}) x)

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

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

= 15 y (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} x) x)

Simplify

15 y (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} x) x) : 15 y (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} x^{2})

= 15 y (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} x^{2})

\int \frac{x}{( 16 + x ^{2} ) ^{\frac{3}{2}}} d x

y = ln (\frac{1 + sin ( x )}{1 - sin ( x )})

\frac{d}{d x} (x (3 x + 1)^{5})

x \to 1 lim (\frac{sin ( x - 1 )}{x})

\int \frac{ln ( 2 x )}{3 x} d x

What is (\partial)/(\partial x)(15xye^{-x^2-y^2}) ?
The answer to (\partial)/(\partial x)(15xye^{-x^2-y^2}) is 15y(e^{-x^2-y^2}-2e^{-x^2-y^2}x^2)