# The Derivative Product Rule
For functions $f$ and $g$ of $x$, the derivative of their product is:
$\frac d {dx}\bigg[f \cdot g\bigg] = \frac {df} {dx} g + f\frac {dg} {dx}$ ^a1ucyf
The product rule can be applied multiple times to determine higher-level derivatives of products of two functions. The [[Binomial Theorem|Pascal's Triangle]] can be used to simplify this process.
$\frac {d^{(n)}} {dx^{(n)}} \bigg[g(x) \cdot h(x)\bigg] = \sum_{i=0}^n \left[\binom n i g^{(i)}(x) \cdot h^{(n-i)}(x) \right]$ ^crr592
See this [BriTheMathGuy](https://www.youtube.com/watch?v=BjDsJUkDsY4) video for more. This 3Blue1Brown [video](https://www.youtube.com/watch?v=YG15m2VwSjA&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr) is also of note.
When trying to determine the derivative of $f(x)g(x)$ 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.