Mathematics Notes
Chapter: Calculus
Topic: Derivatives
1. Introduction
What is a Derivative?
A derivative is a measure of how a function changes as its input changes. It is a
foundational concept in calculus that helps us understand the rate of change of a
quantity, such as speed, slope, or growth rate.
Real-Life Examples:
The velocity of a car is the derivative of its position with respect to time.
The slope of a hill is the derivative of the elevation as you move along the hill.
2. Fundamental Definition
Mathematical Definition:
The derivative of a function f(x)f(x) at a point x=ax = a is defined as:
f′(a)=limh→0f(a+h)−f(a)hf'(a) = \lim_{{h \to 0}} \frac{f(a + h) - f(a)}{h}
This formula calculates the instantaneous rate of change of f(x)f(x) at x=ax = a.
Key Terms:
f(a+h)f(a + h): The function value slightly ahead of aa.
hh: A small increment in xx.
f′(a)f'(a): The slope of the tangent line to the curve at x=ax = a.
3. Geometric Interpretation
The derivative represents the slope of the tangent line to the graph of f(x)f(x) at a given
point.
Visualization:
Imagine a curve on a graph. The derivative at any point gives the angle or steepness of
the tangent line touching the curve at that point.
4. Physical Interpretation
In physics, the derivative describes the rate at which one quantity changes with respect
to another.
Velocity: Rate of change of position (s(t)s(t)).
Acceleration: Rate of change of velocity (v(t)v(t)).
5. Basic Rules of Differentiation
To simplify the process of finding derivatives, we use the following rules:
5.1. Power Rule
Chapter: Calculus
Topic: Derivatives
1. Introduction
What is a Derivative?
A derivative is a measure of how a function changes as its input changes. It is a
foundational concept in calculus that helps us understand the rate of change of a
quantity, such as speed, slope, or growth rate.
Real-Life Examples:
The velocity of a car is the derivative of its position with respect to time.
The slope of a hill is the derivative of the elevation as you move along the hill.
2. Fundamental Definition
Mathematical Definition:
The derivative of a function f(x)f(x) at a point x=ax = a is defined as:
f′(a)=limh→0f(a+h)−f(a)hf'(a) = \lim_{{h \to 0}} \frac{f(a + h) - f(a)}{h}
This formula calculates the instantaneous rate of change of f(x)f(x) at x=ax = a.
Key Terms:
f(a+h)f(a + h): The function value slightly ahead of aa.
hh: A small increment in xx.
f′(a)f'(a): The slope of the tangent line to the curve at x=ax = a.
3. Geometric Interpretation
The derivative represents the slope of the tangent line to the graph of f(x)f(x) at a given
point.
Visualization:
Imagine a curve on a graph. The derivative at any point gives the angle or steepness of
the tangent line touching the curve at that point.
4. Physical Interpretation
In physics, the derivative describes the rate at which one quantity changes with respect
to another.
Velocity: Rate of change of position (s(t)s(t)).
Acceleration: Rate of change of velocity (v(t)v(t)).
5. Basic Rules of Differentiation
To simplify the process of finding derivatives, we use the following rules:
5.1. Power Rule
