What is (\partial)/(\partial x)((7y)/(x^2+7y^2)) ?

The answer to (\partial)/(\partial x)((7y)/(x^2+7y^2)) is -(14yx)/((x^2+7y^2)^2)

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(\partial)/(\partial x)((7y)/(x^2+7y^2))

\frac{\partial}{\partial x} (\frac{7 y}{x ^{2} + 7 y ^{2}})

- \frac{14 y x}{( x ^{2} + 7 y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{7 y}{x ^{2} + 7 y ^{2}})

Treat

y

as a constant

Take the constant out:

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

= 7 y \frac{\partial}{\partial x} (\frac{1}{x ^{2} + 7 y ^{2}})

Apply exponent rule:

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

= 7 y \frac{\partial}{\partial x} ((x^{2} + 7 y^{2})^{- 1})

Apply the chain rule:

- \frac{1}{( x ^{2} + 7 y ^{2} ) ^{2}} \frac{\partial}{\partial x} (x^{2} + 7 y^{2})

= - \frac{1}{( x ^{2} + 7 y ^{2} ) ^{2}} \frac{\partial}{\partial x} (x^{2} + 7 y^{2})

\frac{\partial}{\partial x} (x^{2} + 7 y^{2}) = 2 x

= 7 y (- \frac{1}{( x ^{2} + 7 y ^{2} ) ^{2}} \cdot 2 x)

Simplify

7 y (- \frac{1}{( x ^{2} + 7 y ^{2} ) ^{2}} \cdot 2 x) : - \frac{14 y x}{( x ^{2} + 7 y ^{2} ) ^{2}}

= - \frac{14 y x}{( x ^{2} + 7 y ^{2} ) ^{2}}

x \to 1 lim (\frac{3}{x} - \frac{2}{ln ( x )})

e^{j (2 t + 4)}

\frac{x ^{4}}{3}

\int 5 x^{4} - 1 d x

\frac{d}{d x} (\frac{5}{x ^{19}})

What is (\partial)/(\partial x)((7y)/(x^2+7y^2)) ?
The answer to (\partial)/(\partial x)((7y)/(x^2+7y^2)) is -(14yx)/((x^2+7y^2)^2)