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Unit 3: Derivatives

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    Calculating velocity and changes in velocity are important uses of calculus, but it is far more widespread than that. Calculus is important in all branches of mathematics, science, and engineering, and it is critical to analysis in business and health as well. In this chapter, we explore one of the main tools of calculus, the derivative, and show convenient ways to calculate derivatives. We apply these rules to a variety of functions in this chapter so that we can then explore applications of these techniques.

    • 3.1: Differentiation Rules
      The derivative of a constant function is zero. The derivative of a power function is a function in which the power on x becomes the coefficient of the term and the power on  x in the derivative decreases by 1. The derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative. The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.
    • 3.2: Derivatives as Rates of Change
      In this section we look at some applications of the derivative by focusing on the interpretation of the derivative as the rate of change of a function. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics.
    • 3.3: The Chain Rule
      Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for \(h(x)=f(g(x)),\) \(h′(x)=f′(g(x))g′(x).\) We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them. The chain rule combines with the power rule to form a new rule: If \(h(x)=(g(x))^n\),then \(h′(x)=n(g(x))^{n−1}g′(x)\).
    • 3.4: Implicit Differentiation
      We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations). By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.
    • 3.5: Exponential and Logarithmic Functions
      The exponential function \(y=b^x\) is increasing if \(b>1\) and decreasing if \(0<b<1\). Its domain is \((−∞,∞)\) and its range is \((0,∞)\). The logarithmic function \(y=\log_b(x)\) is the inverse of \(y=b^x\). Its domain is \((0,∞)\) and its range is \((−∞,∞)\). The natural exponential function is \(y=e^x\) and the natural logarithmic function is \(y=\ln x=\log_e x\). Given an exponential function or logarithmic function in base \(a\), we can make a change of base to convert this function to a
    • 3.6: Derivatives of Exponential and Logarithmic Functions
      In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs, exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions.
    • 3.7: Maxima and Minima
      Finding the maximum and minimum values of a function  has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.
    • 3.8: Derivatives and the Shape of a Graph
      Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
    • 3.9: Limits at Infinity, Asymptotes, and Rational Functions
      We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f as x→±∞ . In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function f.
    • 3.10: Applied Optimization Problems
      One common application of calculus is calculating the minimum or maximum value of a function. For example, companies often want to minimize production costs or maximize revenue. In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume. In this section, we show how to set up these types of minimization and maximization problems and solve them by using the tools developed in this chapter.
    • 3.R: Chapter 3 Review Exercises

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