# Timeless Theorems of Mathematics/Product, Quotient, Composition Rules

The product rule, the quotient rule and the composition *(or chain)*
rule are the most fundamental rules or formulas of differential
calculus. For the functions $u(x)$ and $v(x)$, the rules are,

1.  Product Rule: $D_x (uv) = v\cdot D_x u + u\cdot D_x v$
2.  Quotient Rule:
    $D_x (\frac{u}{v}) = \frac{1}{v^2} (v \cdot D_x u - u \cdot D_x v)$
3.  Composition Rule: $D_x v = D_y v\cdot D_x y$

## Proof

### Product Rule

$D_x (uv)$ $=\lim_{{\Delta x \to 0}} \frac {\Delta (uv)}{\Delta x}$
$=\lim_{{\Delta x \to 0}} \frac{u(x+\Delta x)\cdot v(x+\Delta x) - u(x)\cdot v(x)}{\Delta x}$
$=\lim_{{\Delta x \to 0}} \frac{u(x+\Delta x)\cdot v(x+\Delta x) - u(x)\cdot v(x+\Delta x) + u(x)\cdot v(x+\Delta x) - u(x)\cdot v(x)}{\Delta x}$
$= \lim_{{\Delta x \to 0}} \frac{[u(x+\Delta x)-u(x)]\cdot v(x+\Delta x) + u(x)\cdot [v(x+\Delta x)-v(x)]}{\Delta x}$
$=\lim_{{\Delta x \to 0}} \frac{(\Delta u) \cdot v(x+\Delta x) + u(x)\cdot (\Delta v)}{\Delta x}$
$= \lim_{{\Delta x \to 0}}[\frac {\Delta u}{\Delta x}\cdot v(x + \Delta x) + \frac {\Delta v}{\Delta x}\cdot u(x)]$
$= \lim_{{\Delta x \to 0}}[D_x u\cdot v(x + \Delta x) + u\cdot D_x v]$
$=v\cdot D_x u + u\cdot D_x v$

$\therefore D_x (uv) = v\cdot D_x u + u\cdot D_x v$ ***\[Proved\]***

### Quotient Rule

$D_x (\frac{u}{v})$
$=\lim_{{\Delta x \to 0}} \frac {\Delta (\frac {u}{v})}{\Delta x}$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)}{v(x + \Delta x)} - \frac {u(x)}{v(x)}]$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)\cdot v(x) - u(x)\cdot v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)\cdot v(x) -v(x)\cdot u(x) + v(x)\cdot u(x) - u(x)\cdot v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {v(x)\cdot (u(x + \Delta x)-u(x)) + u(x)\cdot (v(x) - v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {v(x)\cdot (\Delta u) + u(x)\cdot (\Delta v)}{v(x)\cdot v(x + \Delta x)}]$
$=\lim_{{\Delta x \to 0}} \frac {\frac {v(x)\cdot (\Delta u)}{\Delta x} + \frac {u(x)\cdot (\Delta v)}{\Delta x}}{v(x)\cdot v(x + \Delta x)}$
$=\lim_{{\Delta x \to 0}} \frac{{v(x) \cdot \frac{\Delta u}{\Delta x} + u(x) \cdot \frac{\Delta v}{\Delta x}}}{{v(x) \cdot v(x + \Delta x)}}$
$=\lim_{{\Delta x \to 0}} \frac{{v(x) \cdot D_x u + u(x) \cdot D_x v}}{{v(x) \cdot v(x + \Delta x)}}$
$=\frac{{v(x) \cdot D_x u + u(x) \cdot D_x 
v}}{{v(x) \cdot v(x)}}$ $=\frac{{v\cdot D_x u + u\cdot D_x v}}{{v^2}}$

$\therefore D_x (\frac{u}{v}) = \frac{v \cdot D_x u - u \cdot D_x v}{v^2}$

### Composition Rule

$D_xv$ $= \lim_{\Delta x\to 0}\frac{\Delta v}{\Delta x}$
$= \lim_{\Delta x\to 0}\frac{\Delta v\cdot\Delta y}{\Delta x\cdot\Delta y}$
$= \lim_{\Delta x\to 0}\frac{\Delta v}{\Delta y}\cdot\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}$
$= \lim_{\Delta y\to 0}\frac{\Delta v}{\Delta y}\cdot\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}$
$= D_yv\cdot D_xy$

$\therefore D_xv = D_yv\cdot D_xy$

