3: Differentiation Rules
- Page ID
- 208280
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 3 transitions from the theoretical definition of a limit to the practical application of differentiation. It establishes the rules and techniques necessary to calculate the rate of change for any algebraic, trigonometric, or transcendental function, providing the mathematical foundation for physics, engineering, and economics.
- 3.1: Basic Differentiation Rules
- To bypass lengthy limits, we use differentiation rules. The Power Rule states \(\frac{d}{dx}(x^n) = nx^{n-1}\), while the Constant Rule shows the derivative of any constant is zero. These rules, along with Sum and Difference rules, allow us to derive polynomials term-by-term. Uniquely, the Natural Exponential Function \(e^x\) is its own derivative, meaning its slope always equals its value.
- 3.2: The Product and Quotient Rules
- The Product Rule states \((fg)' = f'g + g'f\), ensuring both functions' rates of change are accounted for. For ratios, the Quotient Rule follows \(\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}\). These are essential for differentiating rational functions and locating horizontal tangents by solving \(f'(x)=0\). They provide a more efficient path than the limit definition for calculating instantaneous velocity in complex motion.
- 3.3: Derivatives of Trigonometric Functions
- The derivatives of trigonometric functions are foundational for modeling periodic motion. Using the limit definition, we find the derivatives of all six of these functions and the patterns within them. Notably, sine and cosine derivatives follow a four-step repeating cycle, allowing for easy calculation of any higher-order derivative.
- 3.4: The Chain Rule
- The Chain Rule is the final essential rule, used to differentiate composite functions where one function is "inside" another, like \(f(g(x))\). It states \(y' = f'(g(x)) \cdot g'(x)\). This allows us to more easily differentiate any compositions by breaking the problem into manageable "inner" and "outer" steps.
- 3.5: Implicit Differentiation
- Implicit differentiation is a Chain Rule-based technique used when \(y\) cannot be easily isolated as \(y=f(x)\). Instead of solving for \(y\), we differentiate every term in the equation with respect to \(x\). When differentiating terms containing \(y\), we apply the usual rules but multiply by \(y'\) (or \(\frac{dy}{dx}\)) because \(y\) is treated as an implicit function of \(x\).
- 3.6: Derivatives of Inverse Trigonometric and Logarithmic Functions
- This section introduces differentiation of an inverse function. We use this formula to derive the famous derivatives of the six inverse trigonometric functions and the logarithm. A key focus is logarithmic differentiation. This technique simplifies complex products or functions where both the base and exponent are variables (e.g., \(x^x\)) by taking the natural log of both sides before differentiating implicitly.
- 3.7: Rates of Change in the Natural and Social Sciences
- This section explores how derivatives function as rates of change across various disciplines. In physics, the derivative of a position function \(s(t)\) provides the velocity \(v(t)\), while the derivative of velocity provides acceleration \(a(t)\). Speed is defined as the magnitude of velocity, \(|v(t)|\).
- 3.8: Exponential Growth and Decay
- This section focuses on systems where the rate of change of a quantity is directly proportional to the quantity itself. This relationship is expressed by the differential equation: \(\frac{dy}{dt} = ky\). The general solution to this equation is the exponential model \(y(t) = y_0 e^{kt}\), where \(y_0\) is the initial amount at \(t=0\).
- 3.9: Related Rates
- This section studies how to determine the relationship between the rates of change of two or more related quantities that vary over time. By using the Chain Rule, we can differentiate an equation relating these quantities with respect to time (\(t\)) to find how their derivatives (rates) are connected.
- 3.10: Linear Approximations and Differentials
- This section explores how to use derivatives to approximate function values locally using linear functions. Since tangent lines stay close to a curve near the point of tangency, they provide a simpler way to estimate complex values without a calculator.
- 3.11: Hyperbolic Functions
- This section explores the calculus of hyperbolic functions, which are combinations of \(e^x\) and \(e^{-x}\) that behave similarly to trigonometric functions. They are essential for modeling physical phenomena like water waves, elastic vibrations, and especially the catenary—the natural U-shape formed by a hanging cable or chain supported at its ends.