What is (\partial)/(\partial x)(xe^{xy}+2xy+1/x) ?

The answer to (\partial)/(\partial x)(xe^{xy}+2xy+1/x) is e^{yx}+ye^{yx}x+2y-1/(x^2)

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(\partial)/(\partial x)(xe^{xy}+2xy+1/x)

\frac{\partial}{\partial x} (x e^{x y} + 2 x y + \frac{1}{x})

e^{y x} + y e^{y x} x + 2 y - \frac{1}{x ^{2}}

Solution steps

\frac{\partial}{\partial x} (x e^{x y} + 2 x y + \frac{1}{x})

Treat

y

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (x e^{x y}) + \frac{\partial}{\partial x} (2 x y) + \frac{\partial}{\partial x} (\frac{1}{x})

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

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

\frac{\partial}{\partial x} (\frac{1}{x}) = - \frac{1}{x ^{2}}

= e^{y x} + y e^{y x} x + 2 y - \frac{1}{x ^{2}}

f (x) = 2 x^{2} - 5 x + 3, at x = 0

\int \frac{8 + x + x}{x} d x

\frac{d x}{d t} = x sin (t)

x \to 1 - lim (arcsin (x))

\int x^{2} x^{3} + 4 d x

What is (\partial)/(\partial x)(xe^{xy}+2xy+1/x) ?
The answer to (\partial)/(\partial x)(xe^{xy}+2xy+1/x) is e^{yx}+ye^{yx}x+2y-1/(x^2)