BackBusiness Calculus: The Derivative and Tangent Lines
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Section 1.3: The Derivative
Introduction to the Derivative
The derivative of a function provides the slope of the tangent line at any point x on the function's graph. The process of finding the derivative is called differentiation. Understanding derivatives is fundamental in Business Calculus, as they are used to analyze rates of change, optimize functions, and model real-world phenomena.
Derivative: Measures the instantaneous rate of change of a function.
Differentiation: The process of computing the derivative.
Notations for the Derivative
There are several common notations for the derivative of a function y = f(x):
Derivative of Linear and Power Functions
Linear Functions
For a linear function, the derivative is constant and represents the slope of the line.
If , then Explanation: The slope of a straight line is constant for all values of x.
If (a constant function), then Explanation: The slope of a horizontal line is zero.
Power Functions
The derivative of a power function uses the Power Rule:
If , then
Example: If , then
Examples of Differentiation
Example 1: Find the Derivatives
(a)
(b) Rewrite as
(c) Rewrite as
Finding the Equation of the Tangent Line
General Steps
To find the equation of the tangent line to the graph of at the point where :
Step 1: Find the slope of the tangent line
Find the derivative function
Evaluate to get the slope at
Step 2: Find the equation of the tangent line
Evaluate to find the point of contact
Use the point-slope form:
Example: Tangent Line to at
Step 1: Slope at :
Step 2: Point: Equation: Simplified:
Example: Tangent Line to at
Step 1: Slope at :
Step 2: Point: Equation: Simplified:
Summary Table: Derivative Rules
Function | Derivative | Notes |
|---|---|---|
Linear function; slope is constant | ||
Constant function; slope is zero | ||
Power Rule | ||
Rewrite as before differentiating |
Applications in Business Calculus
Derivatives are used to find marginal cost and marginal revenue in economics.
Tangent lines help approximate function values and analyze instantaneous rates of change.
Understanding slopes and rates is essential for optimization problems in business contexts.
Additional info: The examples and explanations have been expanded for clarity and completeness, including step-by-step differentiation and tangent line calculations, as well as a summary table for quick reference.
