The chain rule is a fundamental concept in calculus that plays a crucial role in calculus, particularly in solving derivative problems. Despite its importance, students often struggle with understanding and applying the chain rule.
The truth is that the chain rule is a simple rule that helps us calculate the derivative of composite functions. Sounds complicated? Don’t worry; in this blog post, I will explain the chain rule and how to use it in an easy-to-understand way. Check out this article if you wonder how to find vertical asymptotes of rational functions.
What Is The Chain Rule?
The chain rule is used when finding the derivative of composite functions. A composite function is one where multiple functions are combined together.
To calculate the derivative of a composite function, we need to use the chain rule. The chain rule states that if we have a composite function f(g(x)), the derivative of that function is given by the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function.
The general formula for the chain rule is dy/dx = dy/du × du/dx, in other words, the derivative of f(x) is equal to h'(g(x)) * g'(x).
Here’s an example to help you understand:
Suppose we have the function f(x) = (x2 + 3)4. To find the derivative of this function using the chain rule
- We can first rewrite it as f(g(x)), with g(x) = x2 + 3.
- Next, we find the derivative of g(x), which is 2x.
- Now we can use the chain rule and substitute our findings into the formula:
f'(x) = 4(x2 + 3)3 * 2x
Simplifying this, we get the following:
f'(x) = 8x(x2 + 3)3
Another way of thinking of the chain rule is the following:
- f(x) = Un, with U being any function
- In our example, f(x) = (x2 + 3)4 where U = x2 + 3 and n = 4
- f'(x) = nUn-1 U’
- 4(x2 + 3)4-1 *2x
- There f'(x) = 8x(x2 + 3)
Examples Of Using The Chain Rule
Consider the function f(x) = (2x – 7)4, which can be written as f(x) = h(g(x)) where g(x) = 2x – 7 and h(x) = x4.
To find the derivative of f(x) using the chain rule, we first find the derivatives of g(x) and h(x). The derivative of g(x) is g'(x) = 2, and the derivative of h(x) is h'(x) = 4x3.
Thus, the derivative of f(x) is f'(x) = 8(2x – 7)3.
Using The Chain Rule To Find The Derivative Of Trigonometric Functions
Consider the function f(x) = sin(2x2 – 3x). This function can be broken down into two functions, g(x) = 2x2 – 3x and h(x) = sin(x). The derivative of g(x) is g'(x) = 4x – 3, and the derivative of h(x) is h'(x) = cos(x). Thus, the derivative of f(x) is f'(x) = cos(2x2 – 3x) * (4x – 3).
I encourage you to check out Khan Academy or watch this video or the video below to learn more about the chain rule.
What to read next:
- Slope Intercept Form: Definition, how to find, and its applications.
- Finding the Equation of a Tangent Line with Derivatives.
- Solving Systems of Equations by the Elimination Method.
Wrapping Up
The chain rule is essential in calculus as it helps students to find the derivative of composite functions. As a formula, the chain rule says that the derivative of a composite function is equal to the derivative of the outside function multiplied by the derivative of the inside function.
Understanding the chain rule will make you faster when performing derivatives of complex functions, including trigonometric, logarithm, and exponential functions.