# The Derivative Product Rule $\frac d {dx}\bigg[f(x)g(x)\bigg] = f'(x)g(x) + f(x)g'(x)$ ^a1ucyf This can be remembered using a mnemonic: left d(right) + right d(left) (note that it's reversed to the rule written above). 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 is also of note: https://www.youtube.com/watch?v=YG15m2VwSjA&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr. When trying to determine the derivative of $f(x)g(x)$ think about what it represents, namely the area of a rectangle whose one 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 $ We're trying to determine the derivative though, and not the new area, so we're actually interested in how the area changes with $\Delta x$. $ \frac d {dx} \bigg[f(x)g(x)\bigg] = \frac {\Delta A} {\Delta x} = \frac {\Delta f} {\Delta x} g(x) + f(x) \frac {\Delta g} {\Delta x} + \frac {\Delta f} {\Delta x} \Delta g $ Notice that $\frac {\Delta f} {\Delta x}$ is actually the derivative of $f$, and that $\frac {\Delta g} {\Delta x}$ is the derivative of $g$. We can then simplify the above to $ \frac d {dx} \bigg[f(x)g(x)\bigg] = f'(x) g(x) + f(x) g'(x) + f'(x) \Delta g $ The last part the above is actually zero because when $\Delta x$ goes to zero, $\Delta g$ will go to zero as well. This means we can actually ignore the green rectangle.