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

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

$\therefore D_x(uv)=v\cdot D_{x}u+u\cdot D_{x}v$ [Proved]

Quotient Rule

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

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

Composition Rule

$D_{x}v$ $=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\mathrm{\Delta }v}{\mathrm{\Delta }x}$ $=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\mathrm{\Delta }v\cdot \mathrm{\Delta }y}{\mathrm{\Delta }x\cdot \mathrm{\Delta }y}$ $=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\mathrm{\Delta }v}{\mathrm{\Delta }y}\cdot \underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$ $=\underset{\mathrm{\Delta }y\to 0}{\lim }\frac{\mathrm{\Delta }v}{\mathrm{\Delta }y}\cdot \underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$ $=D_{y}v\cdot D_{x}y$

$\therefore D_{x}v=D_{y}v\cdot D_{x}y$

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