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 […]

The derivative defines the rate at which one variable changes with respect to another.  It is an important concept that comes in extremely useful in many applications: in everyday life, the derivative can tell you at which speed you are driving, or help you predict fluctuations on the stock market; in machine learning, derivatives are important for function optimization.  This tutorial will explore different applications of derivatives, starting with the more familiar ones before moving to machine learning. We will […]

It is often desirable to study functions that depend on many variables.  Multivariate calculus provides us with the tools to do so by extending the concepts that we find in calculus, such as the computation of the rate of change, to multiple variables. It plays an essential role in the process of training a neural network, where the gradient is used extensively to update the model parameters.  In this tutorial, you will discover a gentle introduction to multivariate calculus.  After […]

Integral calculus was one of the greatest discoveries of Newton and Leibniz. Their work independently led to the proof, and recognition of the importance of the fundamental theorem of calculus, which linked integrals to derivatives. With the discovery of integrals, areas and volumes could thereafter be studied.  Integral calculus is the second half of the calculus journey that we will be exploring. In this tutorial, you will discover the relationship between differential and integral calculus.  After completing this tutorial, you […]

Vector valued functions are often encountered in machine learning, computer graphics and computer vision algorithms. They are particularly useful for defining the parametric equations of space curves. It is important to gain a basic understanding of vector valued functions to grasp more complex concepts. In this tutorial, you will discover what vector valued functions are, how to define them and some examples. After completing this tutorial, you will know: Definition of vector valued functions Derivatives of vector valued functions Let’s […]

Partial derivatives and gradient vectors are used very often in machine learning algorithms for finding the minimum or maximum of a function. Gradient vectors are used in the training of neural networks, logistic regression, and many other classification and regression problems. In this tutorial, you will discover partial derivatives and the gradient vector. After completing this tutorial, you will know: Function of several variables Level sets, contours and graphs of a function of two variables Partial derivatives of a function […]

Gradient descent procedure is a method that holds paramount importance in machine learning. It is often used for minimizing error functions in classification and regression problems. It is also used in training neural networks, and deep learning architectures. In this tutorial, you will discover the gradient descent procedure. After completing this tutorial, you will know: Gradient descent method Importance of gradient descent in machine learning Let’s get started. A Gentle Introduction to gradient descent. Photo by Mehreen  

Higher-order derivatives can capture information about a function that first-order derivatives on their own cannot capture.  First-order derivatives can capture important information, such as the rate of change, but on their own they cannot distinguish between local minima or maxima, where the rate of change is zero for both. Several optimization algorithms address this limitation by exploiting the use of higher-order derivatives, such as in Newton’s method where the second-order derivatives are used to reach the local minimum of an […]