“Back in my school days up to the 10th grade, I had a genuine love for math. Whether it was tackling geometry, diving into trigonometry, or exploring progressions, I felt pretty confident in my abilities. But then came derivatives, and suddenly everything took a sharp turn. Instead of visualizing and understanding the beauty of math, I found myself stuck in a maze of formulas and differentiation problem-solving.
For the longest time, I never really questioned what derivatives were all about. Maybe my teachers didn’t delve into the practical side of it, and it always felt like a ritual of proving ‘LHS = RHS.’ As time passed, my relationship with derivatives grew distant.
It wasn’t until the world of computer science brought us back together. As I ventured into machine learning, I couldn’t shake the question: What on earth is a derivative? So, I dived into textbooks, watched countless videos, and finally, it all clicked. It was like seeing equations through a new lens, and it completely changed how I approached and appreciated the math that once seemed elusive.”
What exactly is a Derivative
“Let’s break down the idea of derivatives in a way that makes it easy for anyone who’s been on the same journey of trying to grasp what they are. So, here we go!
Imagine a simple straight line, like y = x. Now, we can rewrite this line as y — x = 0. Picture that line in your mind; it’s pretty straightforward, right? Since it’s a straight line, we can confidently say that the rate at which it changes (what we call slope) is constant. Picture any two points on this line, and you’ll see that the rate of change between those two points remains the same — a concept we often express as (y2 — y1) / (x2 — x1).
Now, shift gears and think about a different equation, like y = x². Look at the graph it creates — it’s not a straight line anymore. The rate of change on this curve isn’t constant; it shifts at every point along the way. Calculating the slope at a specific spot, or the rate of change at a particular point, or the slope of the graph at a given point — all of this is what we call the derivative.
In simpler terms, when you’re dealing with a straight line, the rate of change is steady. But when you’re working with more complex curves, understanding derivatives helps us pinpoint how fast things are changing at any specific moment. It’s like capturing the dynamic essence of a curve, and that’s what we mean when we talk about derivatives in the world of math.”
Alright, let’s simplify this idea about finding the slope at a specific point on a curve. Imagine you’re trying to draw a line that just touches the curve at a single point — we call that line a tangent and we can say the rate of change at that particular point is same as the slope of tangent. Now, it’s not as simple as connecting two random points on the curve, because there are countless lines that can pass through one point.
So, the trick is to find two points on the curve that are incredibly close to each other, like practically right next to each other, but not exactly on top of each other. And this is where the concept of limits comes into play.
Let’s call the tiny distance between these two points “h.” Picture two points on the curve with x-coordinates x and x+h. The y-coordinates of these points are f(x) and f(x+h), respectively. Now, to find the slope, we use the formula:
In simpler terms, we’re talking about making that tiny distance, h, smaller and smaller, approaching zero. This helps us nail down the slope of the tangent at that specific point on the curve. So, it’s like finding the rate of change at an incredibly tiny scale, and that’s the essence of how derivatives work in the world of math. So the derivative of function f(x) is written s f’ (x) and can be written as.
Now lets try to use this formulae in our example y = x².
And that wraps up our journey through derivatives! From navigating the challenges of these mathematical concepts to rediscovering their relevance in the realm of machine learning, we’ve witnessed how derivatives are more than just abstract formulas. They serve as tools that unveil the secrets behind how things change. Whether we’re dealing with a straightforward line or a more intricate curve like y = x², derivatives shine in capturing the nuanced details of how functions transform. So, as we conclude, let’s keep in mind that in the language of mathematics, comprehending change is the key to unraveling the mysteries of the world. Until our paths cross again, happy exploring!
And now Gradient Descent should make more sense for you!!!