# Chain Rule Formula

## Chain Rule Formula

The Chain Rule Formula is used in differential calculus to get the derivative of a composite function. According to the chain rule, if y = f(g(x)), then the instantaneous rate of change of function f with respect to g and g with respect to x leads to an instantaneous rate of change of f with respect to x. As a result, y’ = f'(g(x)) will be supplied as the derivative of y. g'(x). One of the key rules of distinction is the chain rule. With cases that have been solved, the article on the Extramarks website and mobile application teaches the Chain Rule Formula.

## What is Chain Rule?

The Chain Rule Formula is used to determine the derivative of a composite function, such as cos 2x, log 2x, etc. Another name for it is the composite function rule. Only composite functions are subject to the Chain Rule Formula. Therefore, before beginning the Chain Rule Formula, students should first examine what a composite function is and how it may be distinguished.

### Chain Rule Steps

• Step 1: The function must be a composite function, which indicates that one function is nested above another, according to the Chain Rule Formula.
• Step 2: Determine the inner and outside functions.
• Step 3: Without affecting the inner function, find the derivative of the outer function.
• Step 4: to determine the inner function’s derivative.
• Step 5: Add the outcomes of steps 4 and 5.
• Step 6: Condense the derivative of the chain rule.
• For instance: Think of a function: g(x) = ln(sin x) (sin x)
• A composite function is g. Use the chain principle.
• The inner function is called sin, and the outside function is called ln(x).
• 1/sin x is the outer function’s derivative.
• Cos x is the inner function’s derivative.
• The derivative of the inside function is hence g'(x) = derivative of the outer function, leaving the inside function alone, where 1/sin x cos x.
• When students simplify, they will see that cos x/sin x = cot x.

## Chain Rule Formula and Proof

Students can find the Chain Rule Formula and Proof notes and solutions on the Extramarks website and mobile application. These solutions have been designed keeping in mind the requirements of each student. Therefore, students can study without worrying about anything else.

### Chain Rule Formula 1:

Chain Rule Formula 1:

f(g(x)) = f’ (g(x)) g’ (x)

As an illustration, write sin 2x = f(g(x)), where f(x) = sin x and g(x) = 2x, to determine the derivative of d/dx (sin 2x).

The Chain Rule Formula therefore reads: d/dx (sin 2x) = cos 2x 2 = 2 cos 2x.

### Chain Rule Formula 2:

Chain Rule Formula 2:

Students can assume that the expression is applying the Chain Rule Formula and changing “x” to “u.”

Dy/dx equals Dy/Du x Du/Dx

Example: Assume that y = sin 2x and 2x = u in order to calculate d/dx (sin 2x). If so, y = sin u.

According to the chain rule equation, d/dx (sin 2x) = d/du (sin u) Cos u = 2 cos 2x = d/dx(2x) = cos u

### Chain Rule formula Proof

Students should visit the Extramarks website and mobile application to obtain further information on the Chain Rule Formula and its study materials. They can also download the study materials and notes cited on the Extramarks website and mobile application in PDF format and use it for offline study.

## Double Chain Rule

When a function depends on more than one variable, the functions may be layered one on top of the other. The overall derivative is obtained by multiplying a series of smaller derivatives. Three functions, u, v, and w, please. The components of a function, f, are u, v, and w. Here, the chain rule is expanded. The chain rule is used twice when a function is made up of three other functions. When f = (u o v) o, then w = df/dx = df/du/dv, then dv/dw, then dw/dx

## Applications of The Chain Rule

Physics, chemistry, and engineering all make extensive use of the Chain Rule Formula. The chain rule is used:

• To determine the time rate of pressure change,
• To determine how quickly the distance between two moving objects changes,
• To determine the location of a moving object to the right and left in a specific interval,
• How can one tell if a function is growing or shrinking?
• To determine how quickly the average molecular speed is changing.

## Solved Examples on Chain Rule Formula

Example 1: Applying the chain rule, get the derivative of y=lnx.

The answer is y = ln x.

Because f(x) = y is made up of the functions ln(x) and x, we may distinguish it using the chain rule.

Suppose that u = x. If so, y = ln u.