What is (\partial)/(\partial x)(x/(4x^2+9y^2)) ?

The answer to (\partial)/(\partial x)(x/(4x^2+9y^2)) is (-4x^2+9y^2)/((4x^2+9y^2)^2)

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(\partial)/(\partial x)(x/(4x^2+9y^2))

\frac{\partial}{\partial x} (\frac{x}{4 x ^{2} + 9 y ^{2}})

\frac{- 4 x ^{2} + 9 y ^{2}}{( 4 x ^{2} + 9 y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{x}{4 x ^{2} + 9 y ^{2}})

Treat

y

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial x} ( x ) ( 4 x ^{2} + 9 y ^{2} ) - \frac{\partial}{\partial x} ( 4 x ^{2} + 9 y ^{2} ) x}{( 4 x ^{2} + 9 y ^{2} ) ^{2}}

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (4 x^{2} + 9 y^{2}) = 8 x

= \frac{1 \cdot ( 4 x ^{2} + 9 y ^{2} ) - 8 xx}{( 4 x ^{2} + 9 y ^{2} ) ^{2}}

1 \cdot (4 x^{2} + 9 y^{2}) - 8 xx = - 4 x^{2} + 9 y^{2}

= \frac{- 4 x ^{2} + 9 y ^{2}}{( 4 x ^{2} + 9 y ^{2} ) ^{2}}

y = \frac{x + 5}{x ^{3} + x - 9}

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

\int \frac{3 - 18 x}{1 - 9 x ^{2}} d x

\frac{d}{d x} (ln (x - 5))

\frac{d}{d x} (\frac{x + 1}{x ^{3} + x + 1})

What is (\partial)/(\partial x)(x/(4x^2+9y^2)) ?
The answer to (\partial)/(\partial x)(x/(4x^2+9y^2)) is (-4x^2+9y^2)/((4x^2+9y^2)^2)