What is (\partial)/(\partial x)(1/(3x^2+y)) ?

The answer to (\partial)/(\partial x)(1/(3x^2+y)) is -(6x)/((3x^2+y)^2)

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(\partial)/(\partial x)(1/(3x^2+y))

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

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

Solution steps

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

Treat

y

as a constant

Apply exponent rule:

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

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

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (3 x^{2} + y) = 6 x

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

Simplify

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

= - \frac{6 x}{( 3 x ^{2} + y ) ^{2}}

x \to - 3 lim (g (x) (f (x)))

\int x (1 - x)^{10} d x

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

\frac{d}{d x} (2 x lo g_{10} (x))

t \frac{d Q}{d t} + Q = t^{5} ln (t)

What is (\partial)/(\partial x)(1/(3x^2+y)) ?
The answer to (\partial)/(\partial x)(1/(3x^2+y)) is -(6x)/((3x^2+y)^2)