Derivatives and Rate of Change
MAT1241: Calculus1
University of Eswatini
In this comprehensive guide, we’ll explore the fundamental concept of derivatives and their
connection to rates of change. Understanding derivatives is essential for analyzing functions,
finding tangent lines, and optimizing real-world problems. Let’s dive deep into the topic and
provide step-by-step explanations along with examples.
1. Introduction to Derivatives
a. Definition
The derivative of a function (f(x)) at a point (x = a) represents the rate at which the function
changes with respect to (x). It provides information about the slope of the tangent line to the
graph of (f(x)) at that point.
b. Notation
The derivative of (f(x)) with respect to (x) is denoted as:
2. Calculating Derivatives
a. Power Rule
For a function , where (n) is a constant:
Example 1
Find the derivative of .
Solution:
b. Sum and Difference Rules
For functions (u(x)) and (v(x)):
1.
2.
MAT1241: Calculus1
University of Eswatini
In this comprehensive guide, we’ll explore the fundamental concept of derivatives and their
connection to rates of change. Understanding derivatives is essential for analyzing functions,
finding tangent lines, and optimizing real-world problems. Let’s dive deep into the topic and
provide step-by-step explanations along with examples.
1. Introduction to Derivatives
a. Definition
The derivative of a function (f(x)) at a point (x = a) represents the rate at which the function
changes with respect to (x). It provides information about the slope of the tangent line to the
graph of (f(x)) at that point.
b. Notation
The derivative of (f(x)) with respect to (x) is denoted as:
2. Calculating Derivatives
a. Power Rule
For a function , where (n) is a constant:
Example 1
Find the derivative of .
Solution:
b. Sum and Difference Rules
For functions (u(x)) and (v(x)):
1.
2.
