What is (\partial)/(\partial x)(e^{-(x^2+xy+y^2)}) ?

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

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(\partial)/(\partial x)(e^{-(x^2+xy+y^2)})

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

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

Solution steps

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

Simplify

e^{- (x^{2} + x y + y^{2})} : e^{- x^{2} - x y - y^{2}}

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

(2 x y - 3) d x + (x^{2} + 4 y) d y = 0, y (1) = 2

\int 2 x^{2} x^{3} + 1 d x

x \to + 2 lim (\frac{2 x - 25}{x - 2})

f (t) = arcsin (\frac{1}{t})

x \to - 0.1 lim (\frac{sin ( 3 x )}{x})

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