# 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 $\Delta x$. <div> <img src="https://d18l82el6cdm1i.cloudfront.net/uploads/ZGiTfmc9kD-c3q2p2.svg" style="background: white; padding: 20px; width: 600px" /> </div> 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.