Composite Functions and Chain Rule
Let’s say we have a function f(x) = (x + 1)2, for which we want to calculate the derivative. These kinds of functions are called composite functions, which means they are made up of more than one function. Usually, they are of the form g(x) = h(f(x)) or it can also be written as g = hof(x). In our case, the given function f(x) = (x + 1)2 is composed of two functions,
f(x) = g(h(x))
where,
- g(x) = x2
- h(x) = x + 1
For example,
f(x) = (x + 1)2
f(x) = x2 + 1 + 2x
Differentiating the function with respect to x,
f'(x) = 2x + 1
Derivatives of Composite Functions
Derivatives are an essential part of calculus. They help us in calculating the rate of change, maxima, and minima for the functions. Derivatives by definition are given by using limits, which is called the first form of the derivative. We already know how to calculate the derivatives for standard functions, but sometimes we need to deal with complex mathematical functions that are composed of more than two functions. It becomes hard to calculate the derivative for such functions in the normal way. It becomes essential to learn about the rules and methods which make our calculation easier. The chain rule is one of them, which allows us to calculate the derivatives of complex functions.
In this article, we will learn about derivatives of Composite Functions, Examples, and others in detail.
Table of Content
- What is Derivative of Composite Functions?
- Derivatives of Composite Functions Formula
- Composite Functions and Chain Rule
- Chain Rule
- Alternative Method to Chain Rule
- Derivatives of Composite Functions In One Variable
- Derivatives of Composite Functions Examples
- Practice Questions
- FAQs on Derivatives