Differentials, Derivatives, and Applications

- Overview

A derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. It is one of the most critical concepts in the entire field. 

Differentiation is the process of finding the derivative of a function. When x is substituted into the derivative, the result is the slope of the original function y = f (x). 

The slope is often expressed as the “rise” over the “run,” or, in Cartesian terms, the ratio of the change in y to the change in x.

Please refer to the following for more details:

• Wikipedia: Derivative
• Wikipedia: Differential Calculus
• Wikipedia: Tangent

- Derivatives for Machine Learning

In machine learning (ML), derivatives are a fundamental concept in calculus that measure how a function changes as its input varies. They play a crucial role in many ML algorithms. 

In ML, derivatives are often used to: optimize models, minimize errors, refine predictions. 

Derivatives can also be used to: 

• Optimize functions
• Find local minima and maxima
• Calculate gradients for backpropagation in neural networks

Optimization algorithms like gradient descent use derivatives to decide whether to increase or decrease weights. For example, they can be used to maximize or minimize a model's accuracy or error functions. 

While it's not difficult to learn the basics of calculus, not all ML projects require a strong understanding of calculus. For example, some say that the main part of mathematics for ML is statistics.

- Differential Calculus

Differential calculus is a branch of mathematics that studies the rate at which quantities change. It's one of the two traditional divisions of calculus, the other being integral calculus. 

Differential calculus is concerned with the rate of change of one quantity with respect to another. For example, velocity is the rate of change of distance with respect to time in a particular direction. 

Differential calculus starts with a formula for average rate of change, which is essentially a slope calculation. Then, using limits, a formula for the instantaneous rate of change can be developed, which is called the derivative of a function. 

The derivative of a function can often be used to approximate certain function values with a surprising degree of accuracy. 

Differential calculus was devised by Isaac Newton and G.W. Leibniz.

- Differentials vs. Derivatives

Differential and derivative are two fundamental concepts in calculus that are often used interchangeably but are not the same thing. A differential is an infinitesimal change in a variable, while a derivative is a measure of how much the function changes with respect to its input.

[More to come ...]