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 […]

# Archive | Calculus

## A Gentle Introduction to Multivariate Calculus

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 […]

## Applications of Derivatives

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 […]

## A Gentle Introduction to Continuous Functions

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 […]

## A Gentle Introduction to Indeterminate Forms and L’Hospital’s Rule

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 […]

## The Power, Product and Quotient Rules

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. […]

## 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, […]

## 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 […]

## 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 […]

## 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 […]