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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Sum/Difference Rule:

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

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

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

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

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

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

Simplify

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

y = ln (x + 5)

x \to 2 lim (x^{2} + 6)

\frac{( 4 x ^{2} + 4 x + 6 )}{x}

\int \frac{x + 5}{2 x + 3} d x

\int t e^{- 8 t} d t

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