What is (\partial)/(\partial y)(sqrt(2x^2+e^{2y)}) ?

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{2 2 x ^{2} + e ^{2 y}} \cdot 2 e^{2 y} : \frac{e ^{2 y}}{2 x ^{2} + e ^{2 y}}

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

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