Differentiation

From Average to Instantaneous Change

Consider a road trip. Your average speed for the whole trip might be 60 km/h. But at any single moment, your speedometer might read 100 km/h or 0 km/h. The 60 km/h is an average rate of change over a long time interval. The speedometer reading is the instantaneous rate of change. How do we calculate that 'at a moment' value? This is the central question the derivative answers.

The derivative is the mathematical tool for finding the instantaneous rate of change of a function.

The Geometric View: Slope of a Tangent

We can see this on a graph. The average rate of change between two points on a curve is simply the slope of the straight line connecting them (a secant line).

$$\text{Average Slope} = \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}$$

To find the instantaneous rate of change at a single point, we need to make the interval $\Delta x$ incredibly small. Slide the second point closer and closer to the first point. The secant line will pivot, getting closer and closer to becoming the tangent line—the line that just touches the curve at that single point.

The Formal Definition of the Derivative

To capture this idea of making the interval 'infinitely small', we use the concept of a limit. The derivative of a function $f(x)$ with respect to $x$ is defined as:

$$\frac{dy}{dx} = f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$

This fundamental formula is the basis of differential calculus. It takes the expression for the slope of a secant line and finds its limit as the interval size shrinks to zero.

Common Notations

• Leibniz Notation: $\frac{dy}{dx}$ - emphasizes the 'change in y' over 'change in x'.
• Lagrange Notation: $f'(x)$ (read 'f prime of x') - compact and function-focused.
• Newton's Notation: $\dot{x}$ - commonly used in physics specifically for derivatives with respect to time (e.g., velocity).

The Derivative in Physics

The derivative is arguably the most important mathematical tool in physics. It allows us to define many fundamental quantities:

• Velocity: The instantaneous rate of change of position with respect to time.
  $v = \frac{dx}{dt}$
• Acceleration: The instantaneous rate of change of velocity with respect to time.
  $a = \frac{dv}{dt}$
• Force: The rate of change of momentum with respect to time.
  $F = \frac{dp}{dt}$
• Current: The rate of flow of charge with respect to time.
  $I = \frac{dQ}{dt}$

The Differentiation Toolkit

Finding the derivative from the limit definition every time is tedious. Mathematicians have developed a set of powerful, time-saving rules. Mastering these rules is the key to becoming proficient in calculus.

Standard Derivatives Reference Table

Before applying the rules below, it helps to have the derivatives of the most common functions on hand. Each of these can itself be derived from the limit definition, but in practice they are used as a ready reference.

1. The Power Rule

This is the most common rule you will use. It applies to any term of the form $x^n$.

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

In words: Bring the power down in front as a multiplier, then subtract one from the power.

• Example 1: $\frac{d}{dx}(x^5) = 5x^{5-1} = 5x^4$
• Example 2 (Square Root): $\frac{d}{dx}(\sqrt{x}) = \frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$

(Also, the derivative of a constant is 0, and the derivative of a constant times a function is the constant times the derivative of the function.)

2. The Product Rule

Used when you have two functions of x multiplied together, like $f(x) = u(x)v(x)$.

$$\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$$

In words: The first function times the derivative of the second, plus the second function times the derivative of the first.

Example: Differentiate $y = x^2 \sin(x)$.
Here, $u=x^2$ and $v=\sin(x)$.
$\frac{dy}{dx} = x^2 \frac{d}{dx}(\sin(x)) + \sin(x) \frac{d}{dx}(x^2)$
$\frac{dy}{dx} = x^2\cos(x) + 2x\sin(x)$

3. The Quotient Rule

Used when you have one function divided by another, $f(x) = u(x)/v(x)$.

$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$$

In words: 'Low D-high minus High D-low, over the square of what's below.'

Example: Differentiate $y = \frac{x^3}{x^2+1}$.
Here, $u=x^3$ and $v=x^2+1$.
$\frac{dy}{dx} = \frac{(x^2+1)(3x^2) - (x^3)(2x)}{(x^2+1)^2} = \frac{3x^4+3x^2 - 2x^4}{(x^2+1)^2} = \frac{x^4+3x^2}{(x^2+1)^2}$

4. The Chain Rule

This rule is used for 'nested' functions (a function inside of a function). If $y = f(g(x))$, we let $u = g(x)$ so $y = f(u)$.

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

In words: The derivative of the outer function (with the inside function left alone) times the derivative of the inner function.

Example: Differentiate $y = (x^2+1)^3$.
The outer function is $u^3$ and the inner function is $u=x^2+1$.
$\frac{dy}{dx} = (3(x^2+1)^2) \cdot (2x) = 6x(x^2+1)^2$

Putting the Limit Definition to Work

The formal limit definition of the derivative can feel abstract until you use it directly on a real quantity. The following examples build the idea of instantaneous velocity from the ground up, starting with an average rate of change over a finite interval and then shrinking that interval to zero.

Applying the Standard Rules

Once the derivative table and the sum, product, and chain rules are known, most derivatives can be found without ever returning to the limit definition. The following examples show these rules in action.

Differentiating a Derivative

Since the derivative of a function, $f'(x)$, is itself a function, we can take its derivative as well. This is called the second derivative, and we can continue this process to find the third, fourth, and even higher-order derivatives.

Notation

• Second Derivative: $f''(x)$ or $\frac{d^2y}{dx^2}$
• Third Derivative: $f'''(x)$ or $\frac{d^3y}{dx^3}$

The Meaning of the Second Derivative

The second derivative tells us the rate of change of the rate of change. This might sound abstract, but it has two very important and intuitive interpretations.

1. Physical Meaning: Acceleration

In physics, the second derivative is most famously known as acceleration.

• Position, $x(t)$: Where an object is.
• First Derivative, $v(t) = \frac{dx}{dt}$: The rate of change of position (how fast it's moving).
• Second Derivative, $a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}$: The rate of change of velocity (how fast its speed or direction is changing).

A positive acceleration means the velocity is increasing, while a negative acceleration (deceleration) means the velocity is decreasing.

2. Geometric Meaning: Concavity

Geometrically, the second derivative tells us about the concavity of a function's graph—which way the curve is bending.

• If $f''(x) > 0$, the graph is concave up (like a cup holding water). The slope is increasing.
• If $f''(x) < 0$, the graph is concave down (like a cup spilling water). The slope is decreasing.

Example: Position, Velocity, and Acceleration

Suppose an object's position is given by the function $x(t) = t^3 - 6t^2 + 9t + 1$.

1. Find the velocity, $v(t)$:
$v(t) = \frac{dx}{dt} = 3t^2 - 12t + 9$

2. Find the acceleration, $a(t)$:
$a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2} = 6t - 12$

Using these, we can find the velocity and acceleration at any time. For instance, at $t=1$, the object has zero velocity ($v(1)=0$) but a negative acceleration ($a(1)=-6$), meaning it is momentarily at rest but about to move in the negative direction.

The Need for Limits

Calculus is the mathematics of change. But to understand change at a single instant, we first need a tool to describe what happens as we get infinitely close to that instant. This tool is the limit. The concept of a limit is the foundation on which all of calculus is built.

The Idea of a Limit

A limit answers the question: 'What value is a function approaching as its input approaches a certain point?'

Consider the function $f(x) = x^2$. What value does this function 'approach' as $x$ gets closer and closer to 2? It's easy to see that as $x$ gets closer to 2, $f(x)$ gets closer to 4. We write this formally as:

$$\lim_{x \to 2} x^2 = 4$$

This reads: 'The limit of $x^2$ as $x$ approaches 2 is 4.'

Why Not Just Plug In the Number?

In the example above, we could have just plugged in $x=2$ to get 4. But consider this function:

$$g(x) = \frac{x^2 - 4}{x - 2}$$

What is the limit as $x$ approaches 2? If we plug in $x=2$, we get $\frac{0}{0}$, which is undefined. But a limit doesn't care about what happens *at* the point, only what happens as we get *infinitely close* to it. For any $x \neq 2$, we can simplify the function: $g(x) = \frac{(x-2)(x+2)}{x-2} = x+2$. Now, as $x$ gets closer and closer to 2, $g(x)$ gets closer and closer to 4. So, $\lim_{x \to 2} g(x) = 4$, even though the function itself is undefined at that point.

Continuity: An Unbroken Path

Now that we understand limits, we can define continuity. A function is continuous at a point if it meets three conditions:

1. The function is defined at that point (the point exists).
2. The limit of the function exists at that point (the function approaches a single value from both sides).
3. The limit and the function's value are the same.

Intuitively, a continuous function is one whose graph you can draw without lifting your pen from the paper. There are no sudden jumps, holes, or breaks.