What is (\partial)/(\partial x)((5y)/(x^2+1)) ?

The answer to (\partial)/(\partial x)((5y)/(x^2+1)) is -(10yx)/((x^2+1)^2)

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(\partial)/(\partial x)((5y)/(x^2+1))

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

- \frac{10 y x}{( x ^{2} + 1 ) ^{2}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

5 y (- \frac{1}{( x ^{2} + 1 ) ^{2}} \cdot 2 x) : - \frac{10 y x}{( x ^{2} + 1 ) ^{2}}

= - \frac{10 y x}{( x ^{2} + 1 ) ^{2}}

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

\int_{- 3}^{3} (\frac{27}{2} - \frac{3}{2} x^{2}) d x

x \to 0 - lim (\frac{x - 2}{x ^{2} - 4})

\frac{( q x + r )}{( s x + t )}

f (x) = \frac{e ^{x}}{3 x ^{2} + 2 x ^{4}}

What is (\partial)/(\partial x)((5y)/(x^2+1)) ?
The answer to (\partial)/(\partial x)((5y)/(x^2+1)) is -(10yx)/((x^2+1)^2)