# The Derivative Product Rule
For functions $u$ and $v$ of $x$, the derivative of their product is:
$\frac d {dx}\bigg[u \cdot v\bigg] = du \cdot v + u \cdot dv$ ^a1ucyf
The product rule can be applied multiple times using [[Binomial Theorem|Pascal's Triangle]], to easily determine higher-order derivatives of products of two functions. For more see this video: [BriTheMathGuy](https://www.youtube.com/watch?v=BjDsJUkDsY4).
When trying to determine the derivative of $f(x) \cdot g(x)$ think about what it represents, namely the area of a rectangle whose one side is $f(x)$ and the other is $g(x)$. Now think about what happens when you increase $x$ by $\Delta x$.
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The area of the new rectangle $A_2$ corresponds to the area of the old rectangle plus the two yellow rectangles and the green rectangle. Namely:
$A_2 = f(x)g(x) + \Delta f g(x) + f(x) \Delta g + \Delta f \Delta g $
What we're interested in however is not the new area but the change in area, which corresponds to the areas of the yellow and green rectangles:
$\Delta A = \Delta f g(x) + f(x) \Delta g + \Delta f \Delta g$
The derivative then is the change of area as approaches 0:
$\frac d {dx} \bigg[f \cdot g\bigg] = \frac {df} {dx} g + f \frac {dg} {dx} + \frac {df} {dx} \frac {dg} {dx}$
The last part, which corresponds to the area of the green rectangle, approaches 0.