Throughout Chapter 2, we will be working to develop shortcut derivative rules that will help us to bypass the limit definition of the derivative in order to quickly determine the formula for \(f'(x)\) when we are given a formula for \(f (x)\).
- 2.1: Elementary Derivative Rules
- The limit definition of the derivative leads to patterns among certain families of functions that enable us to compute derivative formulas without resorting directly to the limit definition. If we are given a constant multiple of a function whose derivative we know, or a sum of functions whose derivatives we know, the Constant Multiple and Sum Rules make it straightforward to compute the derivative of the overall function.
- 2.2: The Sine and Cosine Function
- In this section, we are going to work to conjecture formulas for the sine and cosine functions, primarily through a graphical argument. To help set the stage for doing so, the following preview activity asks you to think about exponential functions and why it is reasonable to think that the derivative of an exponential function is a constant times the exponential function itself.
- 2.3: The Product and Quotient Rules
- If a function is a sum, product, or quotient of simpler functions, then we can use the sum, product, or quotient rules to differentiate the overall function in terms of the simpler functions and their derivatives. The product and quotient rules now complement the constant multiple and sum rules and enable us to compute the derivative of any function that consists of sums, constant multiples, products, and quotients of basic functions we already know how to differentiate.
- 2.4: Derivatives of Other Trigonometric Functions
- The derivatives of the other four trigonometric functions are derived. These four rules for the derivatives of the tangent, cotangent, secant, and cosecant can be used along with the rules for power functions, exponential functions, and the sine and cosine, as well as the sum, constant multiple, product, and quotient rules, to quickly differentiate a wide range of different functions.
- 2.5: The Chain Rule
- In this section, we encountered the following important ideas: A composite function is one where the input variable x first passes through one function, and then the resulting output passes through another.
- 2.6: Derivatives of Inverse Functions
- Because each function represents a process, a natural question to ask is whether or not the particular process can be reversed. That is, if we know the output that results from the function, can we determine the input that led to it? Connected to this question, we now also ask: if we know how fast a particular process is changing, can we determine how fast the inverse process is changing?
- 2.7: Derivatives of Functions Given Implicitely
- Implicit Differentiation is used to identfy the derivative of a y(x) function from an equation where y cannot be solved for explicitly in terms of x, but where portions of the curve can be thought of as being generated by explicit functions of x. In this case, we say that y is an implicit function of x. The process of implicit differentiation, we take the equation that generates an implicitly given curve and differentiate both sides with respect to x while treating y as a function of x.
- 2.8: Using Derivatives to Evaluate Limits
- Derivatives be used to help us evaluate indeterminate limits of the form 0 0 through L’Hopital’s Rule, which is developed by replacing the functions in the numerator and denominator with their tangent line approximations. A version of L’Hopital’s Rule also allows us to use derivatives to assist us in evaluating other indeterminate limits.
- 2.E: Computing Derivatives (Exercises)
- These are homework exercises to accompany Chapter 2 of Boelkins et al. "Active Calculus" Textmap.
Contributors and Attributions
Matt Boelkins (Grand Valley State University), David Austin (Grand Valley State University), Steve Schlicker (Grand Valley State University)