What is (\partial)/(\partial x)((8y)/(9x^2+4y^2+5)) ?

The answer to (\partial)/(\partial x)((8y)/(9x^2+4y^2+5)) is -(144yx)/((9x^2+4y^2+5)^2)

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(\partial)/(\partial x)((8y)/(9x^2+4y^2+5))

\frac{\partial}{\partial x} (\frac{8 y}{9 x ^{2} + 4 y ^{2} + 5})

- \frac{144 y x}{( 9 x ^{2} + 4 y ^{2} + 5 ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{8 y}{9 x ^{2} + 4 y ^{2} + 5})

Treat

y

as a constant

Take the constant out:

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

= 8 y \frac{\partial}{\partial x} (\frac{1}{9 x ^{2} + 4 y ^{2} + 5})

Apply exponent rule:

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

= 8 y \frac{\partial}{\partial x} ((9 x^{2} + 4 y^{2} + 5)^{- 1})

Apply the chain rule:

- \frac{1}{( 9 x ^{2} + 4 y ^{2} + 5 ) ^{2}} \frac{\partial}{\partial x} (9 x^{2} + 4 y^{2} + 5)

= - \frac{1}{( 9 x ^{2} + 4 y ^{2} + 5 ) ^{2}} \frac{\partial}{\partial x} (9 x^{2} + 4 y^{2} + 5)

\frac{\partial}{\partial x} (9 x^{2} + 4 y^{2} + 5) = 18 x

= 8 y (- \frac{1}{( 9 x ^{2} + 4 y ^{2} + 5 ) ^{2}} \cdot 18 x)

Simplify

8 y (- \frac{1}{( 9 x ^{2} + 4 y ^{2} + 5 ) ^{2}} \cdot 18 x) : - \frac{144 y x}{( 9 x ^{2} + 4 y ^{2} + 5 ) ^{2}}

= - \frac{144 y x}{( 9 x ^{2} + 4 y ^{2} + 5 ) ^{2}}

\frac{d y}{d x} = - sin^{2} (y)

n = 1 \sum \infty (- 1)^{n} (- 1)^{n}

\frac{d}{d x} (5 x^{l n (x)})

\frac{d y}{d x} y = x^{x}

8 x^{3} - 18 x^{2} - 6

What is (\partial)/(\partial x)((8y)/(9x^2+4y^2+5)) ?
The answer to (\partial)/(\partial x)((8y)/(9x^2+4y^2+5)) is -(144yx)/((9x^2+4y^2+5)^2)