What is (\partial)/(\partial x)((3y)/(x^2+2)) ?

The answer to (\partial)/(\partial x)((3y)/(x^2+2)) is -(6yx)/((x^2+2)^2)

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(\partial)/(\partial x)((3y)/(x^2+2))

\frac{\partial}{\partial x} (\frac{3 y}{x ^{2} + 2})

- \frac{6 y x}{( x ^{2} + 2 ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{3 y}{x ^{2} + 2})

Treat

y

as a constant

Take the constant out:

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

= 3 y \frac{\partial}{\partial x} (\frac{1}{x ^{2} + 2})

Apply exponent rule:

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

= 3 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

= 3 y (- \frac{1}{( x ^{2} + 2 ) ^{2}} \cdot 2 x)

Simplify

3 y (- \frac{1}{( x ^{2} + 2 ) ^{2}} \cdot 2 x) : - \frac{6 y x}{( x ^{2} + 2 ) ^{2}}

= - \frac{6 y x}{( x ^{2} + 2 ) ^{2}}

\frac{d}{d x} (\frac{3}{4} x^{\frac{4}{3}} - \frac{3}{8} x^{\frac{2}{3}})

x - x^{3}

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

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

\int \frac{a ^{t}}{t} d t

What is (\partial)/(\partial x)((3y)/(x^2+2)) ?
The answer to (\partial)/(\partial x)((3y)/(x^2+2)) is -(6yx)/((x^2+2)^2)