Differentiation of the product of two functions

Let $y = uv$. By the General Rule,

• FIRST STEP. $y + \Delta y = (u + \Delta u)(v + \Delta v)$. Multiplying out this becomes
  $$y + \Delta y = uv + u \cdot \Delta v + v \cdot \Delta u + \Delta u \cdot \Delta v.$$

• SECOND STEP. $\Delta y = u \cdot \Delta v + v \cdot \Delta u + \Delta u \cdot \Delta v$.
• THIRD STEP. $\frac{\Delta y}{\Delta x} = u \frac{\Delta v}{\Delta x} + v \frac{\Delta u}{\Delta x} + \Delta u \frac{\Delta v}{\Delta x}$.
• FOURTH STEP. $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$. [Applying Theorem 3.8.1], since when $\Delta x \to 0$, $\Delta u \to 0$, and $\left ( \Delta u \frac{\Delta v}{\Delta x} \right ) \to 0$.]

Therefore, $\frac{d}{dx} (uv) = u\frac{dv}{dx}\ +\ v\frac{du}{dx}$ (equation (V) above).

Product rule: The derivative of the product of two functions is equal to the first function times the derivative of the second, plus the second function times the derivative of the first.

Here's how to use Sage to compute an example of this rule:

[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]

sage: t = var("t")
sage: f = cos(t)
sage: g =  exp(2*t)
sage: diff(f*g,t)
2*e^(2*t)*cos(t) - e^(2*t)*sin(t)
sage: diff(f,t)*g+f*diff(g,t)
2*e^(2*t)*cos(t) - e^(2*t)*sin(t)

This simply computes $\frac{d}{dt}(e^{2t}\cos(t)$ in two ways (one: directly, the second: using the product rule) and checks that they are the same.