What is (\partial)/(\partial x)(x^8e^{xy}) ?

The answer to (\partial)/(\partial x)(x^8e^{xy}) is 8x^7e^{xy}+e^{xy}yx^8

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(\partial)/(\partial x)(x^8e^{xy})

\frac{\partial}{\partial x} (x^{8} e^{x y})

8 x^{7} e^{x y} + e^{x y} y x^{8}

Solution steps

\frac{\partial}{\partial x} (x^{8} e^{x y})

Treat

y

as a constant

Apply the Product Rule:

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

f = x^{8}, g = e^{x y}

= \frac{\partial}{\partial x} (x^{8}) e^{x y} + \frac{\partial}{\partial x} (e^{x y}) x^{8}

\frac{\partial}{\partial x} (x^{8}) = 8 x^{7}

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

= 8 x^{7} e^{x y} + e^{x y} y x^{8}

\int 4 x^{2} e^{- 2 x} d x

\frac{\partial}{\partial x} (x^{- 2} \cdot \frac{y}{x ^{3}})

y y^{^{'}} + x y^{2} = x y^{4}

\frac{d}{d x} (\frac{1}{2} x^{- 2})

\int x^{5} I n x d x

What is (\partial)/(\partial x)(x^8e^{xy}) ?
The answer to (\partial)/(\partial x)(x^8e^{xy}) is 8x^7e^{xy}+e^{xy}yx^8