Differentiability : From Approaching and Unbroken to Beyond Change (Part-3)

 Welcome back ! Dear readers,

This has been quite a journey, starting with the concept of 'approaching' in limits, moving on to the 'unbroken' nature of continuous functions, and finally delving into the realm of 'differentiability', where we explore the dynamics of change. 

Have you wondered how fast the world is changing? How fast new data is being generated and changed at the same time? How the world is evolving so fast? Concept of rate of change is a fundamental block of our world. Being a Data Science and Artificial Intelligence student, this amazes me quite often.

History and Philosophy

The development of calculus is often connected with two brilliant minds: Sir Isaac Newton and Gottfried Wilhelm Leibniz. Working separately in the late 17th century, they laid the foundation for this revolutionary field of mathematics. Newton's focus was on understanding the motion of celestial bodies, while Leibniz explored the more general concept of infinitesimal change. Their combined efforts gave us what we call today as calculus.

From a Philosophical perspective change is important, change is how humans have come to where they are... We started as micro-organisms and because of constantly adapting to change we have become such intelligent, capable organisms. This just makes me wonder more about how all of our actions in the current world will affect the world that is to come.

Differentiability, Where did it all come from?

Following is the definition of differentiability,

A function f(x) is said to be differentiable at x = a, if

lim_{x→a} [f(x) - f(a)]  / [x - a] ,  exists.

i.e. f '(a) = lim_{x→a} (f(x) - f(a)) / (x - a)

Since, the differentiation is the limiting value of a function at x = a, both the following limits must exist and be equal

lim_{x→a} (f(a+h) - f(a)) / h        =        lim_{x→a} (f(a-h) - f(a)) / -h

[Hence,  Rf '(a) = Lf '(a) ]

Lf '(a) = LHD = Left hand derivative of f(x) at x = a 

Rf '(a) = RHD = Right hand derivative of f(x) at x = a

• A function is said to be differentiable in (a,b) if it is differentiable at every point in the interval.

• In case of the [a, b], the function should have derivatives at every point and at the end points a & b, R f'(a) = L f'(b).

This definition is where all the differentiation formulae for trigonometric functions came from. This is the base formula and it actually leads to the further things we shall discuss now.

Defining Differentiability

In layman terms, differentiability is about understanding how a function changes at a specific point. Imagine a curve on a graph. The derivative of a function at a point represents the slope of the tangent line to the curve at that point.

Tangent Line: Think of a tangent line as a line that just touches the curve at a single point without crossing it. The slope of this tangent line tells us how steeply the curve is rising or falling at that particular point.

Differentiability and Continuity

There's a crucial connection between differentiability and continuity. A function must be continuous at a point before it can be differentiable at that point.

• Continuity means that the function is unbroken and has no jumps or gaps at that point.

• Differentiability means that the function not only just exists and is continuous at the point, but also has a well-defined slope (or tangent line) at that point.

Think of it like this: if a road has a sudden jump or a sharp corner, you can't smoothly drive over it. Similarly, if a function has a sharp corner or a discontinuity, you can't define a smooth tangent line at that point.

Differentiation Rules

To calculate derivatives, we use a set of rules:

• Power Rule: For functions of the form f(x) = x^n, the derivative is f'(x) = n*x^(n-1).
• Product Rule: Used to find the derivative of the product of two functions.
• Quotient Rule: Used to find the derivative of the ratio of two functions.
• Chain Rule: Used to find the derivative of composite functions (functions within functions).

Real-World Applications of Derivatives

Derivatives have numerous applications in various fields:

• Physics: Velocity is the derivative of position, and acceleration is the derivative of velocity. Derivatives are essential for analyzing the motion of objects, from the trajectory of a projectile to the orbit of a planet.

• Economics: Derivatives are used to calculate marginal cost, marginal revenue, and marginal profit, which are essential concepts in economic decision-making.

• Machine Learning: Derivatives are used to train neural networks, a powerful type of artificial intelligence. The process of training involves adjusting the parameters of the network to minimize errors, and this optimization process relies heavily on derivatives.

• Optimization Problems: Derivatives are invaluable for solving optimization problems. We find the points where the derivative is zero. These points refer to the crests and troughs of the curve, representing the maximum and minimum values of the function.

Conclusion

Differentiability provides us with a powerful lens to understand the dynamics of change in our world. From the motion of celestial bodies to the fluctuations of financial markets, derivatives help us model, predict, analyze and optimize various phenomena. 

This was serendipitous indeed. This exploration of limits, continuity, and differentiability has been a truly enlightening and an emotional journey for me. The philosophy of the same has deeply moved me and how the seemingly abstract concepts are so intertwined with our world has me in awe. Honestly, when i was linking these mathematical concepts with my philosophical understanding of things i grew more and more fond of them. And this makes me wonder if any topic, such dearly researched about, could make you get attached to it the same way or not...

Well that is a question for me to write about soon !

So don't you worry, I will be back, with my thoughts and this blogpost.

Thankyou so much for reading :)