Chapter 2: Discovering Derivatives
- Page ID
- 116556
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
- 2.1: Derivatives of Polynomial Functions
- This section covers how to find the derivatives of polynomial functions. It introduces the basic power rule for differentiation and demonstrates how to apply it to terms of various degrees. The section includes examples of differentiating polynomials and highlights the key steps for finding first and higher-order derivatives of polynomial functions. The focus is on understanding the straightforward process of differentiating terms of the form \(x^n\).
- 2.2: Differentiation Techniques - The Product and Quotient Rules
- This section introduces the Product and Quotient Rules for differentiation. It explains how to differentiate products of two functions using the Product Rule and quotients using the Quotient Rule. Examples demonstrate the step-by-step process for applying these rules to various functions, helping to differentiate more complex expressions involving multiplication and division of functions.
- 2.3: Derivatives of Trigonometric Functions
- This section explains how to differentiate the six basic trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. It covers the rules for each function and provides examples to illustrate how to apply these differentiation techniques. The section also explores the relationships between the derivatives of these functions and how they can be used in various calculus problems.
- 2.4: Differentiation Techniques - The Chain Rule
- This section introduces the chain rule for differentiating composite functions. It explains how to apply the rule, which involves taking the derivative of the outer function while multiplying by the derivative of the inner function. Examples are provided to demonstrate how the chain rule works for various types of composite functions, including trigonometric, exponential, and polynomial functions.
- 2.5: An Interlude - Hyperbolic Functions
- The material in this section is likely not review. Instead, it introduces an important family of functions called the hyperbolic functions. These functions are used throughout calculus and differential equations.
- 2.6: Derivatives of Exponential and Hyperbolic Functions
- This section covers the differentiation of exponential and hyperbolic functions. It explains how to find derivatives of exponential functions, focusing on base \(e\), and explores the rules for differentiating hyperbolic functions like sinh, cosh, and others. The section provides examples to apply these differentiation techniques to various types of functions and shows their connections to real-world applications.
- 2.7: Implicit Differentiation and the Derivatives of the Inverse Trigonometric and Hyperbolic Functions
- This section introduces implicit differentiation, a technique used to find the derivative when a function is not explicitly solved for one variable. It explains how to differentiate both sides of an equation with respect to a given variable, treating other variables as functions. The section includes examples, such as applying implicit differentiation to equations of circles or other implicit relations, to help demonstrate how this method is useful in calculus problems.
- 2.8: Derivatives of Logarithms and Logarithmic Differentiation
- This section covers the derivatives of logarithmic, inverse trigonometric, and inverse hyperbolic functions. It explains how to differentiate these functions, providing specific formulas for each type and illustrating their application with examples. The section also explores real-world applications of these derivatives in various calculus problems.
- 2.9: Related Rates
- This section covers related rates, a method used in calculus to find the rates at which variables change in relation to each other. It explains how to apply the chain rule to differentiate implicitly with respect to time, helping to solve real-world problems involving changing quantities. The section provides examples such as expanding circles, rising water, and moving objects, illustrating how related rates are applied in various scenarios.
- 2.10: Linear Approximations and Differentials
- This section explains linear approximations and differentials, focusing on how to use the tangent line at a point to approximate the value of a function near that point. It introduces the concept of differentials as a way to estimate small changes in a function based on its derivative. Examples are provided to show how linear approximations can be applied to various functions for practical problem-solving.
Thumbnail: Derivatives (CC BY; OpenStax)