My Initial Experience with Derivatives

I vividly recall my first encounter with derivatives during my high school years. It was a summer weekend, and I was tasked with some frustrating homework. To cope, I brewed a cup of coffee and settled under the apple tree in my parents' garden—no apples fell on my head to inspire me, unfortunately.

At that time, math felt tedious and unexciting to me; I had no real understanding of its beauty. While reading about calculating slopes of function graphs and their applications in optimization, I struggled to grasp the theory. However, I possessed a fancy calculator that handled most calculations for me.

My father joined me and quickly glanced at a particularly tricky problem I was wrestling with. In mere seconds, he provided the solution. I was incredulous—how could he solve it so swiftly? I had been programming my calculator and spent ages on it, while he merely looked at it and knew the answer instantly!

When I finally figured it out, I discovered he was right. I was both amazed and a bit shocked—how could that be? This experience ignited my determination to excel in mathematics, prompting me to devour every math book I could find, even those well beyond my current understanding. I began my journey with derivatives and calculus, becoming proficient within a year.

This tale illustrates that even if your math foundation is shaky, you can start learning derivatives and gradually catch up on the basics.

Understanding Slope

For a linear polynomial, the slope is a constant value, independent of the input variable—this is a defining characteristic of a straight line. Calculus expands this concept to include non-linear functions.

Consider the function:

f(x) = x³ - x² + x.

To analyze the slope, we can draw a secant line intersecting the graph at two points. The slope of this secant line is calculated as Δf/Δx. If we choose points (x, f(x)) and (x + Δx, f(x + Δx)), the slope can be expressed as:

As we allow Δx to approach 0, the second point moves closer to the first. By zooming in on the first intersection point, the graph will resemble the secant line closely.

In the limit, as Δx approaches 0, the secant line becomes a tangent line to the graph at the point (x, f(x)). As we zoom in further, the tangent and the graph become indistinguishable.

Thus, we define the slope of f at that point as the limit of the secant's slope as Δx approaches 0, which becomes the derivative of f, and the process of finding this derivative is known as differentiation.

Have you heard the phrase "to go off on a tangent"? This expression derives from mathematics; while you may be close to the graph at a specific point, moving along the tangent takes you farther away from it.

Unlike linear functions, the derivative of a non-linear function varies—it’s a non-constant function itself. We denote the derivative of f at a point x by f´(x), and the formula for calculating it is valid only if the limit exists, which is essential because this definition applies to specific functions known as differentiable functions.

Examples and Properties from First Principles

In this section, we will explore the concept of differentiability through some examples.

For instance, let’s consider the function f(x) = x². Through careful application of the binomial theorem and our definition, we can derive that the derivative of x² is 2x. In fact, this holds true for any real number a ∈ ℝ.

Notation

Calculus was developed independently by Newton and Leibniz, leading to different notational systems. Generally, we adopt Leibniz's notation, even though Newton received most initial credit for the discovery. The d/dx notation signifies an operator that transforms functions into their derivatives.

It's important to note that df/dx resembles a fraction, which is not mere coincidence. The derivative, defined as the limit of Δf/Δx as Δx approaches 0, suggests that df and dx represent infinitesimals, and their ratio reflects the slope of a tangent line at a given point.

Consider calculations involving df/dx. In certain contexts, we can substitute the limit definition to manipulate Δf and Δx as their real quantities, only to take the limit later. This approach provides significant advantages, allowing us to use fraction rules in some cases. However, it is crucial to recognize that this is a mnemonic device rather than rigorously valid mathematics.

Linearity

In this section, we will demonstrate a key property of the differential operator: its linearity.

Thus, the derivative of a sum equals the sum of the derivatives, and the derivative of a scalar function is the scalar multiplied by the derivative.

To prove this, let h(x) = f(x) + g(x), and similarly for a scalar function c.

Differentiation Rules

This section will cover essential differentiation rules that are fundamental to calculus and should be memorized.

L'Hôpital's Rule

Named after the French mathematician Guillaume de l'Hôpital, this powerful theorem is invaluable for evaluating limits.

Before we present its statement and proof, recall that the definition of the derivative can be expressed equivalently. The rule asserts that if the limits of f(x)/g(x) and g´(x)/f´(x) exist as x approaches some number c, then...

Proof:

Let’s prove a specific yet useful case of this theorem. Suppose f and g are continuously differentiable at a real number c, where f(c) = g(c) = 0, and g´(c) ≠ 0.

Taylor Series

If you are unfamiliar with series, don’t worry; we will cover that in a future article. For now, you can enjoy the exploration of calculus, even if some concepts feel challenging.

One of the most powerful tools in both real and complex analysis is the understanding that analytic functions can be expressed as power series. If f is infinitely differentiable near a point x=a, then within a neighborhood of a, we can express it as...

The notation f^(n) represents the function differentiated n times.

Although I won’t prove this here, differentiating both sides or representing f as a sum can help establish this concept. A rigorous proof will require considering boundedness conditions of the derivatives and convergence of the series.

Applications of Taylor Series

Taylor series are among the most frequently employed tools in calculus. They serve a fundamental role in complex analysis, where a function is either not differentiable or infinitely differentiable, thus allowing for a power series representation.

Let’s leverage these series to affirm the identity of the differential operator. First, we recognize that the exponential function can be expressed as a converging power series.

We can also utilize Taylor series to define trigonometric functions for complex numbers. By writing out the Taylor series for e^x, sine, and cosine, we arrive at...

Ultimately, rearranging these expressions leads us to Euler's identity, which is frequently used to convert between polar and Cartesian coordinates.

This concludes the third chapter in our calculus series. Previous articles can be found here:

In this video titled "Calculus 1 - Derivatives," viewers will gain a foundational understanding of derivatives and their significance in calculus.

The second video, "Derivatives... What? (NancyPi)," provides an engaging exploration of derivatives, making complex concepts accessible to learners.