🚀 Level 1 - Topic 5: Introduction to the Concept of a Derivative 🌟

1) Introduction: The Need for Instantaneous Rate of Change

In our everyday experience, we often deal with rates of change. For instance, we talk about speed (change in distance over time), growth rate of a population, or the slope of a hill. These are often average rates of change over an interval. But calculus introduces us to a more refined idea: instantaneous rate of change.

Consider a car journey. Knowing the average speed for the entire trip is useful, but it doesn't tell us how fast the car was going at any specific moment (e.g., when you sped up to overtake or slowed down for traffic). The derivative is designed to capture precisely this: the rate of change at a single, specific instant.

This concept of instantaneous change is fundamental not just in physics (velocity, acceleration) but also in economics (marginal cost, marginal revenue), engineering (reaction rates, stress in materials), and many other fields. It allows us to analyze how things change at a particular moment, providing much more detailed and powerful insights than average rates.

From Average to Instantaneous Rate: A Limit Process

The derivative is formally defined using the concept of limits. It’s essentially the limit of average rates of change as the interval over which we are averaging shrinks down to zero, focusing on a single point in time or space. This transition from average to instantaneous is what makes derivatives so powerful in calculus.

2) Defining the Derivative: Slope of the Tangent Line

To understand the derivative geometrically, let's think about the graph of a function \( y = f(x) \). The derivative at a point \( x = a \) is intimately connected to the tangent line to the curve at the point \( (a, f(a)) \).

Definition: The Derivative \( f'(a) \)

The derivative of \( f(x) \) at \( x = a \), denoted by \( f'(a) \), is defined as the slope of the tangent line to the curve \( y = f(x) \) at the point \( (a, f(a)) \). This slope represents the instantaneous rate of change of \( f(x) \) with respect to \( x \) at \( x = a \).

The value \( f'(a) \) is found by calculating the limit:

\( f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \)

This limit, if it exists, is the derivative of \( f \) at \( x = a \). Here, \( h \) represents a tiny change in \( x \), and \( f(a + h) - f(a) \) is the corresponding change in \( f(x) \). The ratio \( \frac{f(a + h) - f(a)}{h} \) is the slope of the secant line between \( (a, f(a)) \) and \( (a + h, f(a + h)) \). As \( h \) approaches 0, this secant line approaches the tangent line.

Understanding the Derivative Formula

The formula for the derivative might look abstract at first, but it's deeply intuitive:

\( f(a + h) - f(a) \): This is the "rise" or change in \( y \) values as we move from \( x = a \) to \( x = a + h \).
\( h \): This is the "run" or change in \( x \) values.
\( \frac{f(a + h) - f(a)}{h} \): This is the slope of the secant line. It's an *average* rate of change over the interval from \( a \) to \( a + h \).
\( \lim_{h \to 0} \): By taking the limit as \( h \) approaches zero, we are making the interval infinitesimally small, zooming in on the point \( x = a \), and finding the slope of the tangent line, which is the *instantaneous* rate of change at \( x = a \).

3) Derivative as a Function: \( f'(x) \)

The derivative we defined above, \( f'(a) \), is the derivative at a specific point \( x = a \). However, we can also think of the derivative as a function, \( f'(x) \), that gives us the derivative value for *any* \( x \) in the domain where the derivative exists.

To find the derivative function \( f'(x) \), we simply apply the same limit definition, but instead of using a specific point \( a \), we use a variable \( x \):

\( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \)

If this limit exists for each \( x \) in an interval, then \( f'(x) \) is the derivative function of \( f(x) \) on that interval. The domain of \( f'(x) \) is the set of all \( x \) values for which this limit exists.

4) Rule Box: Notations for the Derivative

Notations for the Derivative

There are several common notations for the derivative, each with its own historical and conceptual emphasis. It's important to be familiar with them all:

\( f'(x) \): "f prime of x". Lagrange's notation. Emphasizes that the derivative is a function derived from \( f \).
\( \frac{dy}{dx} \): "dee y by dee x". Leibniz's notation. Emphasizes the ratio of infinitesimally small changes in \( y \) and \( x \). Useful for remembering units and in related rates problems. If \( y = f(x) \), then \( \frac{dy}{dx} = f'(x) \).
\( \frac{d}{dx} f(x) \): "the derivative of f(x) with respect to x". Emphasizes the operation of differentiation being applied to the function \( f(x) \).
\( y' \): "y prime". Short form when \( y \) is already defined as a function of \( x \).
\( D_x f(x) \): "D sub x of f of x". Operator notation, using \( D_x \) as the differentiation operator.

We will use these notations interchangeably as we progress in calculus. For now, \( f'(x) \) and \( \frac{dy}{dx} \) are most common for introductory purposes.

5) Example Box: Finding the Derivative of a Linear Function

Example 1: Finding the Derivative of a Linear Function

Let's find the derivative of \( f(x) = mx + b \), where \( m \) and \( b \) are constants (slope and y-intercept of a line). We'll use the limit definition of the derivative.

\( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \)

Substitute \( f(x) = mx + b \): \( f'(x) = \lim_{h \to 0} \frac{[m(x + h) + b] - [mx + b]}{h} \)

Simplify: \( f'(x) = \lim_{h \to 0} \frac{mx + mh + b - mx - b}{h} = \lim_{h \to 0} \frac{mh}{h} \)

Cancel \( h \) (for \( h \neq 0 \)): \( f'(x) = \lim_{h \to 0} m = m \)

Therefore, the derivative of \( f(x) = mx + b \) is \( f'(x) = m \). This makes perfect sense! The slope of a linear function is constant everywhere and is equal to \( m \). So, the instantaneous rate of change of a linear function is always its slope.

If \( f(x) = mx + b \), then \( f'(x) = m \). Or, \( \frac{d}{dx}(mx + b) = m \).

6) Example Box: Finding the Derivative of \( f(x) = x^2 \)

Example 2: Finding the Derivative of \( f(x) = x^2 \)

Let's find the derivative of \( f(x) = x^2 \).

\( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} \)

Expand \( (x + h)^2 = x^2 + 2xh + h^2 \): \( f'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2) - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} \)

Factor out \( h \) from the numerator: \( f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} \)

Cancel \( h \) (for \( h \neq 0 \)): \( f'(x) = \lim_{h \to 0} (2x + h) \)

Now take the limit as \( h \to 0 \): \( f'(x) = 2x + 0 = 2x \)

Therefore, the derivative of \( f(x) = x^2 \) is \( f'(x) = 2x \). This means the slope of the tangent line to the parabola \( y = x^2 \) at any point \( x \) is given by \( 2x \). For instance, at \( x = 3 \), the slope is \( f'(3) = 2(3) = 6 \).

If \( f(x) = x^2 \), then \( f'(x) = 2x \). Or, \( \frac{d}{dx}(x^2) = 2x \).

7) Example Box: Finding the Derivative of a Constant Function \( f(x) = c \)

Example 3: Finding the Derivative of a Constant Function \( f(x) = c \)

Let \( f(x) = c \) be a constant function. Let's find its derivative.

\( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \)

Since \( f(x) = c \) for any \( x \), then \( f(x + h) = c \) as well. Substitute into the definition: \( f'(x) = \lim_{h \to 0} \frac{c - c}{h} = \lim_{h \to 0} \frac{0}{h} \)

For any \( h \neq 0 \), \( \frac{0}{h} = 0 \). Therefore: \( f'(x) = \lim_{h \to 0} 0 = 0 \)

The derivative of any constant function is 0. This is because a constant function has no change in its value as \( x \) changes; its rate of change is always zero.

If \( f(x) = c \), then \( f'(x) = 0 \). Or, \( \frac{d}{dx}(c) = 0 \).

8) Example Box: Finding the Derivative of \( f(x) = \frac{1}{x} \)

Example 4: Finding the Derivative of \( f(x) = \frac{1}{x} \)

Let's find the derivative of \( f(x) = \frac{1}{x} \).

\( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h} \)

Combine the fractions in the numerator using a common denominator \( x(x + h) \): \( f'(x) = \lim_{h \to 0} \frac{\frac{x - (x + h)}{x(x + h)}}{h} = \lim_{h \to 0} \frac{\frac{-h}{x(x + h)}}{h} \)

Simplify by dividing by \( h \) (which is the same as multiplying by \( \frac{1}{h} \)): \( f'(x) = \lim_{h \to 0} \frac{-h}{x(x + h)} \cdot \frac{1}{h} = \lim_{h \to 0} \frac{-1}{x(x + h)} \)

Now take the limit as \( h \to 0 \): \( f'(x) = \frac{-1}{x(x + 0)} = \frac{-1}{x^2} \)

Therefore, the derivative of \( f(x) = \frac{1}{x} \) is \( f'(x) = -\frac{1}{x^2} \).

If \( f(x) = \frac{1}{x} \), then \( f'(x) = -\frac{1}{x^2} \). Or, \( \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} \).

9) Note Box: Differentiability and Continuity

Differentiability and Continuity

A crucial fact in calculus is the relationship between differentiability (having a derivative) and continuity:

If a function \( f(x) \) is differentiable at \( x = a \), then it must also be continuous at \( x = a \).

However, the converse is not always true. A function can be continuous at a point but not differentiable there. Think of functions with "corners" or "sharp points" (like \( f(x) = |x| \) at \( x = 0 \)). These functions are continuous but don't have a well-defined tangent line (and thus no derivative) at the sharp point.

Differentiability is a stronger condition than continuity. For a derivative to exist, a function needs to be "smooth" enough – no breaks, jumps, holes, or sharp corners.

10) Summary Box: Key Ideas about Derivatives

Summary: Key Ideas about Derivatives

The derivative \( f'(x) \) of a function \( f(x) \) represents its instantaneous rate of change.
Geometrically, \( f'(a) \) is the slope of the tangent line to the graph of \( f(x) \) at \( x = a \).
The derivative is formally defined as a limit: \( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \).
There are various notations for the derivative (\( f'(x), \frac{dy}{dx}, \frac{d}{dx}f(x) \), etc.).
A differentiable function must be continuous, but a continuous function is not necessarily differentiable (e.g., at sharp corners).

11) Practice Questions - Grasping the Derivative Concept

Test your understanding of the derivative concept with these practice problems.

Fundamental Practice Questions

Instructions: For each function, use the limit definition of the derivative to find \( f'(x) \).

Q1. \( f(x) = 5x - 2 \)
Q2. \( f(x) = 3x^2 + 1 \)
Q3. \( f(x) = -2x + 7 \)
Q4. \( f(x) = x^2 - 4x + 3 \)
Q5. \( f(x) = c \) (where \( c \) is a constant)
Q6. \( f(x) = x^3 \) (Challenge: algebraic expansion needed)
Q7. \( f(x) = 2x^2 - x + 5 \)
Q8. \( f(x) = -4x + 9 \)
Q9. \( f(x) = \frac{1}{2}x - 6 \)
Q10. \( f(x) = ax + b \) (general linear function, constants \( a, b \))
Q11. \( f(x) = x^2 + 2x \)
Q12. \( f(x) = 4 - x^2 \)

Challenging Practice Questions

Instructions: These questions require deeper thinking about derivatives and their properties.

Q1. For the function \( f(x) = x^2 \), find the equation of the tangent line to the graph at the point \( x = 2 \). (Hint: you've already found \( f'(x) \) in Example 2. You need the point and the slope).
Q2. Consider the function \( g(x) = |x| \). Explain graphically why \( g(x) \) is not differentiable at \( x = 0 \). Use the limit definition to show that \( \lim_{h \to 0} \frac{|0 + h| - |0|}{h} \) does not exist. (Consider left-hand and right-hand limits).
Q3. If the position of a particle at time \( t \) is given by \( s(t) = t^2 + 3t \), find the instantaneous velocity of the particle at \( t = 4 \). (Recognize velocity as the derivative of position). What are the units of velocity if position is in meters and time is in seconds?
Q4. Suppose \( f(x) \) is differentiable at \( x = a \). Does \( f(x) \) have to be continuous at \( x = a \)? Explain why or why not. (Refer to the note on differentiability and continuity). If \( f(x) \) is continuous at \( x=a \), must it be differentiable at \( x=a \)? Explain.
Q5. Think of a real-world scenario where the concept of a derivative (instantaneous rate of change) is crucial. Describe the scenario, the function involved (even if abstractly), and what the derivative would represent in that context. (Think beyond just velocity and speed – consider rates of change in economics, biology, etc.). Give an example in economics or biology.

12) Summary & Cheat Sheet - Derivatives at a Glance

Let's summarize the core concepts about derivatives for quick review.

5.1) Definition of the Derivative \( f'(x) \)

\( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \) (if the limit exists).

\( f'(a) \) is the slope of the tangent line to \( y = f(x) \) at \( x = a \) and the instantaneous rate of change of \( f(x) \) at \( x = a \).

5.2) Notations for the Derivative

Common notations include: \( f'(x), \frac{dy}{dx}, \frac{d}{dx}f(x), y', D_x f(x) \).

5.3) Differentiability vs. Continuity

Differentiability implies Continuity: If \( f \) is differentiable at \( a \), then \( f \) is continuous at \( a \). The converse is not always true.