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 […]

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 […]

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 […]

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 […]

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 […]

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 […]

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 […]

Optimization, as one of the core processes in many machine learning algorithms, relies on the use of derivatives in order to decide in which manner to update a model’s parameter values, to maximize or minimize an objective function.  This tutorial will continue exploring the different techniques by which we can find the derivatives of functions. In particular, we will be exploring the power, product and quotient rules, which we can use to arrive to the derivatives of functions faster than […]

Indeterminate forms are often encountered when evaluating limits of functions, and limits in turn play an important role in mathematics and calculus. They are essential for learning about derivatives, gradients, Hessians, and a lot more. In this tutorial, you will discover how to evaluate the limits of indeterminate forms and the L’Hospital’s rule for solving them. After completing this tutorial, you will know: How to evaluate the limits of functions having indeterminate types of the form 0/0 and ∞/∞ L’Hospital’s […]

Many areas of calculus require an understanding of continuous functions. The characteristics of continuous functions, and the study of points of discontinuity are of great interest to the mathematical community. Because of their important properties, continuous functions have practical applications in machine learning algorithms and optimization methods. In this tutorial, you will discover what continuous functions are, their properties, and two important theorems in the study of optimization algorithms, i.e., intermediate value theorem and extreme value theorem. After completing this […]