Now let’s investigate what happens if we, otherwise, had to compute the derivatives of the functions separately first and then multiply them afterwards: In order to investigate how to go about finding the derivative of f( x), let’s first start with finding the derivative of the product of u( x) and v( x) directly: Suppose that we now have a function, f( x), of which we would like to find the derivative, which is the product of another two functions, u( x) = 2 x 2 and v( x) = x 3: We can also apply it to radical functions by first expressing their exponent (or power) as a fraction: The power rule can be applied to any power, be it positive, negative, or a fraction. But if you forget both of these ideas, you can always use the power rule. Or you can just memorize that the derivative of x is 1. The slope (m) of this line is 1, so the derivative equals 1. The best way to understand this derivative is to realize that f(x) = x is a line that fits the form y = mx + b because f(x) = x is the same as f(x) = 1x + 0 (or y = 1x + 0). That is because, when we bring a value of 1 in front of x, and then subtract the power by 1, what we are left with is a value of 0 in the exponent. If applied to f( x) = x, the power rule give us a value of 1. Here, we substitute for f( x) = x 2 and then proceed to simplify the expression:Īs h approaches a value of 0, then this limit approaches 2 x, which tallies with the result that we have obtained earlier using the power rule. If we have a variable base raised to a fixed power, the rule to follow in order to find its derivative is to bring down the power in front of the variable base, and then subtract the power by 1.įor example, if we have the function, f( x) = x 2, of which we would like to find the derivative, we first bring down 2 in front of x and then reduce the power by 1:įor the purpose of understanding better where this rule comes from, let’s take the longer route and find the derivative of f( x) by starting from the definition of a derivative: This tutorial is divided into three parts they are: Photo by Andreas M, some rights reserved. How the quotient rule allows us to find the derivative of a function that is the ratio of two differentiable functions.How the product rule allows us to find the derivative of a function that is defined as the product of another two (or more) functions.

The power rule to follow when finding the derivative of a variable base, raised to a fixed power.In this tutorial, you will discover the power, product and quotient rules to find the derivative of functions.Īfter completing this tutorial, you will know: Hence, for functions that are especially challenging, keeping such rules at hand to find their derivatives will become increasingly important. 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 if we had to find every derivative from first principles. This tutorial will continue exploring the different techniques by which we can find the derivatives of functions. 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.

Quotient and product rule calculus