What is (\partial)/(\partial x)(xe^{-x^2-y^2-z^2}) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(\partial)/(\partial x)(xe^{-x^2-y^2-z^2})

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

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

Solution steps

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

Treat

y, z

as constants

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\int 4 sin (2 θ) d θ

\int_{- 2}^{2} x^{2} - 4 d x

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

f (x) = x + \frac{1}{x}, at x = 4

\int (3 t + 5) cos (\frac{t}{4}) d t

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