What is the Chain Rule in Calculus?
Introduction
The chain rule is a mathematical principle that states that a derivative of a function can be found by applying the derivative of the function to each of the derivatives of the functions that are involved in the chain. This principle is often used to calculate derivatives of complicated functions. This article will help you to learn more about chain rule and its applications.
Definition
It is a method of finding the derivative of a function that is a function of another function. It helps to evaluate the differentiation of a function that can be expressed as a function of another or a function formed by combining two functions.
The chain is applicable for finding derivatives of composite functions only. It is because other differentiation rules are not applicable for a complex or combination of two functions. The product rule is differentiating a function that formed by the product of two different functions.
What is the Chain Rule Formula?
Suppose that we have two functions f(x) and g(x) then the function of f as the function g can be written as:
$$y\;=\;f(g(x))$$
Substitute u = g(x) then we can write above function as:
$$y\;=\;f(u)$$
The chain rule formula is defined as:
$$\frac{dy}{dx}\;=\;\frac{dy}{du}\;\times\;\frac{du}{dx}$$
Where,
- dy / du = is the derivative of y = fu with respect to u.
- du / dx = is the derivative of u = gx with respect to x.
The online chain rule calculator also follow the same formula and provides results quickly.
Derivation of Chain Rule Formula
The chain rule defines the derivative of two functions that are in composition form. Suppose we have two functions f(x) and g(x). The composition of both functions can be written as;
$$y\;=\;f(g(x))\;and\;u\;=\;g(x)$$
Now applying derivatives on both sides with respect to x.
$$\frac{dy}{dx}\;=\;\frac{d}{dx}[f(g(x))]$$
$$\frac{dy}{dx}\;=\;f'(g(x))g'(x)$$
We can write it as,
$$\frac{dy}{dx}\;=\;\frac{dy}{du}\;\times\;\frac{du}{dx}$$
The chain rule formula shows us that we can find the derivative of complicated functions like in a combination of two functions or involve higher order power of polynomials. Let’s discuss the derivation of chain rule in an example.
Example
Find the derivative of f(x) = (3x+1)5
Since the given function has a higher order power, we can easily compute its derivative with chain rule.
Let,
$$u\;=\;3x\;+\;1$$
And the given function can be written as,
$$y\;=\;u^5$$
Applying derivative with respect to u,
$$\frac{dy}{du}\;=\;\frac{d}{du}(u^5)\;=\;5u^{5-1}(1)$$
$$\frac{dy}{du}\;=\;5u^4$$
Now, applying derivative on u with respect to x,
$$\frac{du}{dx}\;=\;\frac{d}{dx}(3x\;+\;1)\;=\;3(1)\;=\;3$$
Since the chain rule is,
$$\frac{dy}{dx}\;=\;\frac{dy}{du}\;\times\;\frac{du}{dx}$$
Substituting the values of derivatives,
$$\frac{dy}{dx}\;=\;5u^4\;x\;3$$
Using the value of u, we get,
$$\frac{dy}{dx}\;=\;5(3x\;+\;1)^4\;×\;3\;=\;15(3x+1)^4$$
Which is the derivative of the given function.
What is the Reverse Chain Rule?
The chain rule for derivatives can be used to calculate derivatives for complex functions that have one or more basic functions. There are many tricks and methods that simplify chain rule for integration in these situations.
The reverse-chain rule can be used for certain classes of functions. This rule can also be called the “substitution Rule", or the “U-Substitution Rule". The reverse chain rule has following form,
$$∫\;f(g(x))g'(x)\;dx\;=\;∫\;f(u)du$$
Where,
$$u\;=\;g(x)$$
The reverse chain rule gives us the integration of composition of two or more functions.
Application of Chain Rule
The chain rule has broad applications in calculus because it gives us a way to calculate the derivative of a composition of functions. Therefore, it is known as a key concept of differentiation.
There are several examples of applications of chain rule because it helps to find derivatives of complex functions. For example, it helps to find implicit, logarithmic, and inverse differentiations.
If we summarise, we can say that the chain rule helps in any problem where the rate of change involves, i.e., change in temperature with time and heat. Besides calculus, it also has many broad applications in physics, chemistry, and engineering. It is used in studying related rates in many disciplines.
In conclusion, the chain rule can be used to find derivatives of functions in composition. This is an important tool for many calculus problems and can be applied to many situations. So remember the chain rule, and you'll be able to solve more complex problems with ease!
FAQ’s
What is the Chain Rule in Differentiation?
The chain rule is used to calculate a function's derivatives where the function's composition is involved. It also helps to find the derivative of a function that is in complex form, for example, a function with a higher order degree. In other words, we can use the definite chain rule as the technique for finding the derivative of a function with respect to another function.
What is the Chain Rule Equation?
For a function y = f(g(x)), the chain rule equation is,
$$\frac{dy}{dx}\;=\;\frac{dy}{du}\;\times\;\frac{du}{dx}$$
Where,
- u = g(x)
- dy / dx = derivative of y with respect to x
- dy / du = derivative of y with respect to u
- du / dx = derivative of u with respect to x
What is the Chain Rule of Derivative?
The chain rule of derivative is a method of finding rate of change of a function with respect to another function, if the function is in a composition form. It is used to calculate derivatives of different functions like, trigonometric, hyperbolic, exponential etc.