Related Article:
Notes:
The constant rule is not included in this infographic. However, the constant rule can be derived from the power rule:
$$\frac{d}{dx}C = \frac{d}{dx}Cx^0 = C \cdot 0x^{-1} = 0.$$.
The constant multiple and sum and difference rules are not included, but linearity combines these two rules. In terms of differentiation, linearity involves the two criteria:
$$\frac{d}{dx}(cf(x) = c\frac{d}{dx}f(x)$$
$$\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}.g(x).$$
In the infographic, we combine these into the equivalent criterion
$$\frac{d}{dx}(af(x) + bg(x)) = a\frac{d}{dx}f(x) + b\frac{d}{dx}g(x).$$
As an example of using the flow chart to compute derivatives, let’s calculate (\((\sin{x} + \cos{x})^2)’\). We have a composition, so we use the chain rule, which gives \(2(\sin{x} + \cos{x})(\sin{x} + \cos{x})’\). Our expression still contains a derivative, so we continue the process. This time we have a linear function, so we apply linearity, arriving at \(2(\sin{x} + \cos{x})(\cos{x} – \sin{x})\). There are no more derivatives, so we are done.
License:
This infographic is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives license. That means you’re free to share and distribute the infographics if you provide proper attribution, don’t use it for commercial purposes, and don’t modify it.
If your intended use falls outside of these rights,, please contact me at legal@themathperimeter.com. We can make arrangements for proper licensing.