What is (\partial)/(\partial x)(xyze^{x^2+y^2+z^2}) ?

The answer to (\partial)/(\partial x)(xyze^{x^2+y^2+z^2}) is yz(e^{x^2+y^2+z^2}+2e^{x^2+y^2+z^2}x^2)

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(\partial)/(\partial x)(xyze^{x^2+y^2+z^2})

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

yz (e^{x^{2} + y^{2} + z^{2}} + 2 e^{x^{2} + y^{2} + z^{2}} x^{2})

Solution steps

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

Treat

y, z

as constants

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

Simplify

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

= yz (e^{x^{2} + y^{2} + z^{2}} + 2 e^{x^{2} + y^{2} + z^{2}} x^{2})

What is (\partial)/(\partial x)(xyze^{x^2+y^2+z^2}) ?
The answer to (\partial)/(\partial x)(xyze^{x^2+y^2+z^2}) is yz(e^{x^2+y^2+z^2}+2e^{x^2+y^2+z^2}x^2)