2: Derivatives

Overview

The derivative is one of the central ideas in calculus. It provides a precise way to describe how a quantity changes at an instant. If \( y = f(x) \), then the derivative \( f'(x) \) measures the instantaneous rate of change of \( f \) with respect to \( x \).

Geometrically, \( f'(x) \) is the slope of the tangent line to the graph of \( y=f(x) \) at \( x \). Physically, derivatives connect to velocity, acceleration, growth rates, and many other real-world rates of change.

The derivative is defined via a limit:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]
In this chapter, we move from the definition to practical rules and applications that make differentiation both powerful and broadly useful.

Topics in this Chapter

1. The Definition of the Derivative
   • The formal limit definition of \( f'(x) \).
   • Interpreted as the instantaneous rate of change and tangent line slope.
2. The Derivative as a Function
   • Understanding the properties of the derivative function.
   • Studying how \(f'(x)\) and \(f(x) \) are related.
3. Differentiation Rules
   • Rules for computing derivatives: constant rule, power rule, product rule, and quotient rule.
4. Derivatives as Rates of Change
   • Interpreting \( f'(x) \) in applied contexts such as velocity, acceleration, and growth.
5. Derivatives of Trigonometric Functions
   • Compute and apply in different contexts the derivative of functions involving trigonometric functions.
6. The Chain Rule
   • Find the derivative of composite functions.
7. Derivatives of Inverse Functions
   • Compute the derivative of functions involving inverse trigonometric functions.
8. Implicit Differentiation
   • Compute derivatives of functions given implicitly, and apply them in different contexts.
9. Derivatives of Exponential and Logarithmic Functions
   • Compute the derivative of functions involving exponential or logarithmic functions, and use them to solve real-life problems.

Why this matters

• Derivatives quantify instantaneous change across science, engineering, and economics.
• They help analyze graphs through slopes, monotonicity, and concavity.
• They are the foundation for optimization, curve sketching, and modeling real-world processes.