Differentiation Infographic

Related Article:

How to Differentiate a Function Step by Step: A Beginner’s Guide

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Chain Rule for Beginners

Learn Implicit Differentiation Fast with This Example

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Master Higher-Order Derivatives

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.

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