What is (\partial)/(\partial x)(xy^2+ye^{x^2}+5) ?

The answer to (\partial)/(\partial x)(xy^2+ye^{x^2}+5) is y^2+2e^{x^2}xy

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(\partial)/(\partial x)(xy^2+ye^{x^2}+5)

\frac{\partial}{\partial x} (x y^{2} + y e^{x^{2}} + 5)

y^{2} + 2 e^{x^{2}} x y

Solution steps

\frac{\partial}{\partial x} (x y^{2} + y e^{x^{2}} + 5)

Treat

y

as a constant

Apply the Sum/Difference Rule:

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

= \frac{\partial}{\partial x} (x y^{2}) + \frac{\partial}{\partial x} (y e^{x^{2}}) + \frac{\partial}{\partial x} (5)

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

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

\frac{\partial}{\partial x} (5) = 0

= y^{2} + y e^{x^{2}} \cdot 2 x + 0

Simplify

= y^{2} + 2 e^{x^{2}} x y

\frac{d y}{d x} + 2 x y = 2 x^{3}, y (0) = 2

\int 1 d ln (x) d x

\frac{d y}{d x} + y = x y^{2}

y = ln (t^{2}) - arctan (\frac{t}{3})

\frac{t + 1710}{50 ( t + 2 )}

What is (\partial)/(\partial x)(xy^2+ye^{x^2}+5) ?
The answer to (\partial)/(\partial x)(xy^2+ye^{x^2}+5) is y^2+2e^{x^2}xy