# What is Calculus?

Calculus is the mathematical study of change.  The effectiveness of calculus to solve a complicated but continuous problem lies in its ability to slice the problem into infinitely simpler parts, solve them separately, and subsequently rebuild them into the original whole. This strategy can be applied to study all continuous elements that can be sliced in this manner, be it the curvatures of geometric shapes, as well as the trajectory of an object in flight, or a time interval.  In […]

# Key Concepts in Calculus: Rate of Change

The measurement of the rate of change is an integral concept in differential calculus, which concerns the mathematics of change and infinitesimals. It allows us to find the relationship between two changing variables and how these affect one another. The measurement of the rate of change is also essential for machine learning, such as in applying gradient descent as the optimisation algorithm to train a neural network model. In this tutorial, you will discover the rate of change as one […]

# Calculus in Machine Learning: Why it Works

Calculus is one of the core mathematical concepts in machine learning that permits us to understand the internal workings of different machine learning algorithms.  One of the important applications of calculus in machine learning is the gradient descent algorithm, which, in tandem with backpropagation, allows us to train a neural network model.  In this tutorial, you will discover the integral role of calculus in machine learning.  After completing this tutorial, you will know: Calculus plays an integral role in understanding […]

# What you need to know before you get started: A brief tour of Calculus Pre-Requisites

We have previously seen that calculus is one of the core mathematical concepts in machine learning that permits us to understand the internal workings of different machine learning algorithms.  Calculus, in turn, builds on several fundamental concepts that derive from algebra and geometry. The importance of having these fundamentals at hand will become even more important as we work our way through more advanced topics of calculus, such as the evaluation of limits and the computation of derivatives, to name […]

# A Gentle Introduction to Limits and Continuity

There is no denying that calculus is a difficult subject. However, if you learn the fundamentals, you will not only be able to grasp the more complex concepts but also find them fascinating. To understand machine learning algorithms, you need to understand concepts such as gradient of a function, Hessians of a matrix, and optimization, etc. The concept of limits and continuity serves as a foundation for all these topics. In this post, you will discover how to evaluate the […]

# A Gentle Introduction to Evaluating Limits

The concept of the limit of a function dates back to Greek scholars such as Eudoxus and Archimedes. While they never formally defined limits, many of their calculations were based upon this concept. Isaac Newton formally defined the notion of a limit and Cauchy refined this idea. Limits form the basis of calculus, which in turn defines the foundation of many machine learning algorithms. Hence, it is important to understand how limits of different types of functions are evaluated. In […]

# A Gentle Introduction to Function Derivatives

The concept of the derivative is the building block of many topics of calculus. It is important for understanding integrals, gradients, Hessians, and much more. In this tutorial, you will discover the definition of a derivative, its notation and how you can compute the derivative based upon this definition. You will also discover why the derivative of a function is a function itself. After completing this tutorial, you will know: The definition of the derivative of a function How to […]

# A Gentle Introduction to Derivatives of Powers and Polynomials

One of the most frequently used functions in machine learning and data science algorithms are polynomials or functions involving powers of x. It is therefore, important to understand how the derivatives of such functions are calculated. In this tutorial, you will discover how to compute the derivative of powers of x and polynomials. After completing this tutorial, you will know: General rule for computing the derivative of polynomials General rule for finding the derivative of a function that involves any […]

# A Gentle Introduction to Slopes and Tangents

The slope of a line, and its relationship to the tangent line of a curve is a fundamental concept in calculus. It is important for a general understanding of function derivatives. In this tutorial, you will discover what is the slope of a line and what is a tangent to a curve. After completing this tutorial, you will know: The slope of a line The average rate of change of f(x) on an interval w.r.t. x The slope of a […]

# Derivative of the Sine and Cosine

Many machine learning algorithms involve an optimization process for different purposes. Optimization refers to the problem of minimizing or maximizing an objective function by altering the value of its inputs.  Optimization algorithms rely on the use of derivatives in order to understand how to alter (increase or decrease) the input values to the objective function, in order to minimize or maximize it. It is, therefore, important that the objective function under consideration is differentiable.  The two fundamental trigonometric functions, the […]