What is (\partial)/(\partial x)((2y)/(x^2+2)) ?

The answer to (\partial)/(\partial x)((2y)/(x^2+2)) is -(4yx)/((x^2+2)^2)

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(\partial)/(\partial x)((2y)/(x^2+2))

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

- \frac{4 y x}{( x ^{2} + 2 ) ^{2}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

2 y (- \frac{1}{( x ^{2} + 2 ) ^{2}} \cdot 2 x) : - \frac{4 y x}{( x ^{2} + 2 ) ^{2}}

= - \frac{4 y x}{( x ^{2} + 2 ) ^{2}}

y = 3 t + 4 t^{3}

\int \frac{x}{( 4 x ^{2} + 3 ) ^{2}} d x

\int cos (2 x) cos (4 x) d x

\frac{d}{d x} (f (h, x) \cdot g (h, x) \cdot h)

\frac{d}{d x} (\frac{1}{5 x})

What is (\partial)/(\partial x)((2y)/(x^2+2)) ?
The answer to (\partial)/(\partial x)((2y)/(x^2+2)) is -(4yx)/((x^2+2)^2)