Lie derivatives: ##L_Xf=[X,f]##The product rule Definition/Summary - What is it?

[Greg Bernhardt]

Definition/Summary

The product rule is a method for finding the derivative of a product of functions.

Equations

$$(fg)'\ =\ f'g\ +\ fg'$$

$$(fgh)'\ =\ f'gh\ +\ fg'h\ +\ fgh'$$

Extended explanation

If a function F is the product of two other functions f and g (i.e. F(x) = f(x)g(x)), then the product rules states that:

$$\frac{d}{dx}F \ = \ \frac{df}{dx}g \ + \ f\frac{dg}{dx}$$

Proof:

$$\frac{F(x + h) - F(x)}{h} \ = \ \frac{f(x + h)g(x + h) - f(x)g(x)}{h}$$

$$= \ \frac{f(x + h)g(x + h) \ - \ f(x)g(x + h) \ + \ f(x)g(x + h) \ - \ f(x)g(x)}{h}$$

$$= \ \frac{f(x + h) - f(x)}{h}g(x + h)<br /> \ + \ f(x)\frac{g(x+h) - g(x)}{h}$$

Now take the limit as h approaches zero.

$$\frac{d}{dx}F \ = \ \lim_{h \to 0}\frac{F(x + h) - F(x)}{h} <br /> \ = \ \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}g(x + h)<br /> \ + \ \lim_{h \to 0}f(x)\frac{g(x+h) - g(x)}{h} <br /> \ = \ \frac{df}{dx}g \ + \ f\frac{dg}{dx}$$

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[fresh_42]

The product or Leibniz rule appears in many variations:

• definition of a derivation: $D(ab)=D(a)b+aD(b)\text{ and }D([a,b)]=[D(a),b]+[a,D(b)]$
• integration by parts: $\int u'v = uv - \int uv'$
• quotient rule: $(f/g)' = (f\cdot g^{-1})'=f\,'g{-1}+(g^{-1})' f$
• Jacobi identity: $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$
• exterior derivatives: $d(f\alpha)=d(f\wedge \alpha)=df\wedge \alpha+f\wedge d\alpha$