What is (\partial)/(\partial y)(sqrt(x+e^{4y)}) ?

The answer to (\partial)/(\partial y)(sqrt(x+e^{4y)}) is (2e^{4y})/(sqrt(x+e^{4y))}

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(\partial)/(\partial y)(sqrt(x+e^{4y)})

\frac{\partial}{\partial y} (x + e^{4 y})

\frac{2 e ^{4 y}}{x + e ^{4 y}}

Solution steps

\frac{\partial}{\partial y} (x + e^{4 y})

Treat

x

as a constant

Apply the chain rule:

\frac{1}{2 x + e ^{4 y}} \frac{\partial}{\partial y} (x + e^{4 y})

= \frac{1}{2 x + e ^{4 y}} \frac{\partial}{\partial y} (x + e^{4 y})

\frac{\partial}{\partial y} (x + e^{4 y}) = 4 e^{4 y}

= \frac{1}{2 x + e ^{4 y}} \cdot 4 e^{4 y}

Simplify

\frac{1}{2 x + e ^{4 y}} \cdot 4 e^{4 y} : \frac{2 e ^{4 y}}{x + e ^{4 y}}

= \frac{2 e ^{4 y}}{x + e ^{4 y}}

\frac{d}{d x} ((2 - x) (y - 2) (3 - x - y))

x \to \infty lim (\frac{x}{8})

x \to 0 + lim (\frac{x ^{2}}{3} - \frac{3}{x})

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

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

What is (\partial)/(\partial y)(sqrt(x+e^{4y)}) ?
The answer to (\partial)/(\partial y)(sqrt(x+e^{4y)}) is (2e^{4y})/(sqrt(x+e^{4y))}