What is (\partial)/(\partial x)(e^xyx+ye^x+y^2ex^x) ?

The answer to (\partial)/(\partial x)(e^xyx+ye^x+y^2ex^x) is e^xxy+2e^xy+ey^2x^xln(x)+ey^2x^x

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(\partial)/(\partial x)(e^xyx+ye^x+y^2ex^x)

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

e^{x} x y + 2 e^{x} y + e y^{2} x^{x} ln (x) + e y^{2} x^{x}

Solution steps

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

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

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

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

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

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

= y (e^{x} x + e^{x}) + y e^{x} + e y^{2} x^{x} (ln (x) + 1)

Simplify

y (e^{x} x + e^{x}) + y e^{x} + e y^{2} x^{x} (ln (x) + 1) : e^{x} x y + 2 e^{x} y + e y^{2} x^{x} ln (x) + e y^{2} x^{x}

= e^{x} x y + 2 e^{x} y + e y^{2} x^{x} ln (x) + e y^{2} x^{x}

x \to 1 lim ((1 - 2 x)^{\frac{1}{x}})

f (x) = \frac{t ^{3} + 3 t}{t ^{3 - 4 t + 3}}

\frac{\partial}{\partial x} (3 sec^{2} (2 x + 3 y))

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

x \to 0 lim (\frac{sin ( x ) - 2 x}{x})

What is (\partial)/(\partial x)(e^xyx+ye^x+y^2ex^x) ?
The answer to (\partial)/(\partial x)(e^xyx+ye^x+y^2ex^x) is e^xxy+2e^xy+ey^2x^xln(x)+ey^2x^x