What is the derivative of 1/(sqrt(xy)) ?

The derivative of 1/(sqrt(xy)) is -(y+x\frac{dy)/(dx)}{2(xy)^{3/2}}

What is the first derivative of 1/(sqrt(xy)) ?

The first derivative of 1/(sqrt(xy)) is -(y+x\frac{dy)/(dx)}{2(xy)^{3/2}}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

\frac{d}{d x} (\frac{1}{x y})

- \frac{y + x \frac{d y}{d x}}{2 ( x y ) ^{\frac{3}{2}}}

Solution steps

\frac{d}{d x} (\frac{1}{x y})

Apply exponent rule:

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

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

Apply the chain rule:

- \frac{1}{2 ( x y ) ^{\frac{3}{2}}} \frac{d}{d x} (x y)

= - \frac{1}{2 ( x y ) ^{\frac{3}{2}}} \frac{d}{d x} (x y)

\frac{d}{d x} (x y) = y + x \frac{d y}{d x}

= - \frac{1}{2 ( x y ) ^{\frac{3}{2}}} (y + x \frac{d y}{d x})

Simplify

- \frac{1}{2 ( x y ) ^{\frac{3}{2}}} (y + x \frac{d y}{d x}) : - \frac{y + x \frac{d y}{d x}}{2 ( x y ) ^{\frac{3}{2}}}

= - \frac{y + x \frac{d y}{d x}}{2 ( x y ) ^{\frac{3}{2}}}

(x^{2} - x y) \frac{d y}{d x} + y^{2} = 0

y = t

\int \frac{2 x - 1}{x ^{2}} d x

3 x^{3} - x^{2} + 9 x - 9

\frac{s - 2}{s ^{2} - s - 6}

What is the derivative of 1/(sqrt(xy)) ?
The derivative of 1/(sqrt(xy)) is -(y+x\frac{dy)/(dx)}{2(xy)^{3/2}}
What is the first derivative of 1/(sqrt(xy)) ?
The first derivative of 1/(sqrt(xy)) is -(y+x\frac{dy)/(dx)}{2(xy)^{3/2}}