Zooming in on the Curve: Discovering Differentiability (Part 3/3, CIA Submission)

A Retrospect to Begin with...

In calculus, the concepts of limits and continuity form the foundation upon which differentiability is built. A limit helps us understand the behavior of a function as it approaches a specific point, while continuity ensures that a function has no abrupt jumps or breaks. Differentiability takes these ideas further, exploring how a function changes at an infinitesimal level.

What is Differentiability?

Differentiability is essentially defined as the ability to derive the function. A derivative refers to the slope of the tangent line to a curve defined at a point on the curve, or alternatively, it can define the function's instantaneous rate of change.
To understand this better, consider the car analogy. If you travel 100 kilometers within 2 hours, the average speed can be calculated as 50 km per hour. Nevertheless, this does not tell whether the car was going really fast or with less speed or was really slow. The derivative describes this aspect by showing the exact speed of the car at a given moment.

Zooming In: The Tangent Line

Now, let’s zoom in on that curve. If you zoom in close enough, the curve will start to look like a straight line. This straight line is called the tangent line, and its slope is what we call the derivative.

If the curve is smooth, the tangent line is easy to define (as seen in the image attached above). But if there’s a sharp corner in the curve, like at the tip of a “V,” no single straight line can describe it. That’s why we say the curve isn’t differentiable at that point (as seen in the image attached below).

How Is Differentiability Different from Continuity?

Continuity is, therefore, a movement that flows smooth and unbroken. That is to say, when you walk along a path without jumping or stopping, your movement is continuous.

Then, one may argue, it is not always easy to consider something as continuous as differentiable. Walking in a smooth path suddenly takes a sharp turn. You would not be stopped (therefore, still continuous), but a sharp corner would mean that your direction changed abruptly, which would not allow you to define your exact "rate of change" at that point.

Real-Life Examples of Differentiability

Differentiability isn’t just a theoretical concept—it’s a practical tool that shows up in many aspects of daily life, science, and technology. Let’s take a closer look at how it’s applied in the real world:

• Measuring Growth: Monitoring Progress of Plants

Imagine you’re growing a sunflower and tracking its height each day. While you can measure how tall it is at the start and end of the week to calculate its average growth, what if you wanted to know how fast it was growing on Wednesday? That’s where differentiability comes in.

By using derivatives, scientists can determine the rate of growth of a plant at any given moment. This is especially useful in agriculture and biology, where understanding growth patterns can help optimize farming techniques or study environmental effects on plants.

• Tracking Speeds: Rockets, Cars and Races

Differentiability is inextricably linked to the study of motion. Just as you would launch a rocket, the speed of a rocket increases as it rises rather than being constant. In order to determine the instantaneous velocity (speed at a specific moment) and acceleration (how quickly its speed changes), engineers use derivatives. 

Now, imagine a Formula 1 car racing around a track. Unlike a regular car, an F1 car undergoes rapid changes in speed and direction as it navigates sharp corners and straightaways. To ensure peak performance, engineers and analysts use derivatives to measure the car’s instantaneous speed, its acceleration, and even the rate at which it decelerates during braking. This data helps teams fine-tune the car's setup and strategy, giving drivers an edge on the track.

Left Hand Derivative and Right Hand Derivative

When studying differentiability, one of the key aspects is ensuring that the derivative exists at a particular point. For this to happen, the function's behavior must be smooth and consistent as we approach the point from both sides. This is where the concepts of the left-hand derivative and the right-hand derivative come into play.

• Left-Hand Derivative (L f ' (x) or LHD): This is the derivative of the function as we approach a point from values smaller than $x$.
• Right-Hand Derivative (R f ' (x) or RHD): This is the derivative of the function as we approach a point from values larger than $x$.

If the left-hand derivative and the right-hand derivative are not equal at a point, the function is not differentiable at that point.

Below are the representation of differentiability, the formulae of LHD and RHD and an example graph attached below for explaining LHD and RHD;

A practical example of analyzing the same is skateboarding. Imagine a skateboarder gliding smoothly down a ramp and reaching the flat ground. As they descend the curved ramp, their motion is smooth, and the slope of their path changes gradually. However, the moment they transition from the curved ramp to the flat surface, there’s an abrupt change in the slope of their path.

As the skateboarder moves down the curved ramp, the slope gradually decreases and becomes flatter. At the bottom of the ramp, the slope approaches a small negative value. As the skateboarder transitions to the flat ground, the slope of their path instantly becomes zero—the flat ground has no incline. At the point where the ramp meets the flat ground, the skateboarder experiences an abrupt change:

• The left-hand slope is negative (indicating a downward incline).
• The right-hand slope is zero (indicating a flat surface).

Since these two slopes are not the same, the motion path is not differentiable at this point. This discontinuity is what the skateboarder physically feels as a sudden "jerk" when moving from the ramp to the flat ground. 

Why Differentiability Matters?

Differentiability is everywhere. It helps engineers design smoother roads, businesses maximize profits, and even athletes optimize their performance. It’s the math behind self-driving cars calculating their speed or economists predicting market trends.

In essence, differentiability gives us a magnifying glass to look closely at change—whether it’s the speed of a car, the growth of a plant, or the rise and fall of the stock market.

Conclusion

Differentiability is more than a mathematical abstraction; it’s a gateway to understanding and predicting change in the real world. From tracking the speed of a car to modeling the spread of diseases, derivatives provide profound insights into dynamic processes. As a cornerstone of calculus, differentiability not only bridges limits and continuity but also opens doors to advanced mathematical techniques that power science, engineering, and beyond.

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Footnote: The materials used in this blog, especially the textual definitions and references are taken from Thomas' Calculus textbook, Calculus textbook by Cengage and my class notes, along with images from various sources from Google.

And that is all for this blogpost on the interesting topic of "Differentiability" for my CIA Assignment submission, assigned by my Mathematics teacher, Dr. Ansa Mathew. Until next time folks, when I arrive with another anecdote of my adventures in Christ University!

 

- Divyabhanu Rana 

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