What is (\partial)/(\partial y)(ye^{-xy}) ?

The answer to (\partial)/(\partial y)(ye^{-xy}) is e^{-yx}-e^{-yx}yx

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(\partial)/(\partial y)(ye^{-xy})

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

e^{- y x} - e^{- y x} y x

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

\frac{d}{d x} (ln (\frac{e ^{x}}{1 + e ^{x}}))

\frac{\partial}{\partial x} (ln ((cos (x) y)^{z}))

x \to 3 lim (3 x + 7)

\int_{0}^{2 π} cos (t) d t

x \to 5 lim (5 - x)

What is (\partial)/(\partial y)(ye^{-xy}) ?
The answer to (\partial)/(\partial y)(ye^{-xy}) is e^{-yx}-e^{-yx}yx