2: Derivatives
- Page ID
- 187910
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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
- The Definition of the Derivative
- The formal limit definition of \( f'(x) \).
- Interpreted as the instantaneous rate of change and tangent line slope.
- The Derivative as a Function
- Understanding the properties of the derivative function.
- Studying how \(f'(x)\) and \(f(x) \) are related.
- Differentiation Rules
- Rules for computing derivatives: constant rule, power rule, product rule, and quotient rule.
- Derivatives as Rates of Change
- Interpreting \( f'(x) \) in applied contexts such as velocity, acceleration, and growth.
- Derivatives of Trigonometric Functions
- Compute and apply in different contexts the derivative of functions involving trigonometric functions.
- The Chain Rule
- Find the derivative of composite functions.
- Derivatives of Inverse Functions
- Compute the derivative of functions involving inverse trigonometric functions.
- Implicit Differentiation
- Compute derivatives of functions given implicitly, and apply them in different contexts.
- 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.