What is (\partial)/(\partial x)(3xe^{-3xy}) ?

The answer to (\partial)/(\partial x)(3xe^{-3xy}) is 3(e^{-3xy}-3e^{-3xy}xy)

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(\partial)/(\partial x)(3xe^{-3xy})

\frac{\partial}{\partial x} (3 x e^{- 3 x y})

3 (e^{- 3 x y} - 3 e^{- 3 x y} x y)

Solution steps

\frac{\partial}{\partial x} (3 x e^{- 3 x y})

Treat

y

as a constant

Take the constant out:

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

= 3 \frac{\partial}{\partial x} (x e^{- 3 x y})

Apply the Product Rule:

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

f = x, g = e^{- 3 x y}

= 3 (\frac{\partial}{\partial x} (x) e^{- 3 x y} + \frac{\partial}{\partial x} (e^{- 3 x y}) x)

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

\frac{\partial}{\partial x} (e^{- 3 x y}) = - 3 y e^{- 3 y x}

= 3 (1 \cdot e^{- 3 x y} + (- 3 y e^{- 3 y x}) x)

Simplify

= 3 (e^{- 3 x y} - 3 e^{- 3 x y} x y)

\int 5 (2 x - 1)^{4} d x

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\int \frac{- 8 x ^{4} + 3 x ^{2} + 9}{3 x ^{3}} d x

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

What is (\partial)/(\partial x)(3xe^{-3xy}) ?
The answer to (\partial)/(\partial x)(3xe^{-3xy}) is 3(e^{-3xy}-3e^{-3xy}xy)