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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial x} (x^{3} + 4 x^{2} y + 4 x + 2)

x \to 2 - lim (\frac{2}{2 - x})

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

(1 + w) w^{3} + 9

y = \frac{5 x}{x}

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