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: $$D_x v = D_y v\cdot D_x y$$

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

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

Quotient Rule
$$D_x (\frac{u}{v})$$ $$=\lim_ \frac {\Delta (\frac {u}{v})}{\Delta x}$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)}{v(x + \Delta x)} - \frac {u(x)}{v(x)}]$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)\cdot v(x) - u(x)\cdot v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)\cdot v(x) -v(x)\cdot u(x) + v(x)\cdot u(x) - u(x)\cdot v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {v(x)\cdot (u(x + \Delta x)-u(x)) + u(x)\cdot (v(x) - v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {v(x)\cdot (\Delta u) + u(x)\cdot (\Delta v)}{v(x)\cdot v(x + \Delta x)}]$$ $$=\lim_ \frac {\frac {v(x)\cdot (\Delta u)}{\Delta x} + \frac {u(x)\cdot (\Delta v)}{\Delta x}}{v(x)\cdot v(x + \Delta x)}$$ $$=\lim_ \frac$$ $$=\lim_ \frac$$ $$=\frac$$ $$=\frac$$

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

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

$$\therefore D_xv = D_yv\cdot D_xy$$