Chain Rule

What is Differentiation?

Differentiation is used to find rates of change. For example, Differentiation allows us to find the rate of change of velocity with respect to time (which gives us acceleration). The concept of differentiation also allows us to find the rate of change of the variable x with respect to variable y, which plotted on a graph of y against x, is known to be the gradient of the curve. Here, in this article we are going to focus on the Chain Rule Differentiation in Mathematics, chain rule examples and chain rule formula example. Let’s define chain rule!

  • The chain rule allows the differentiation of functions that are known to be composite, we can denote chain rule by f∘g, where f and g are two functions. For example, let us take the composite function (x + 3)2. The inner function, namely g equals (x + 3) and if x + 3 = u then the outer function can be written as f = u2.

  • This rule is also known as chain rule because we use it to take derivatives of composites of functions and this happens by chaining together their derivatives.

  • We can think of the chain rule as taking the derivative of the outer function (that is applied to the inner function) and multiplying it times the derivative of the inner function.

\[\frac{d}{dx}\][(f(x))\[^{n}\]] = n(f(x))\[^{n-1}\] . f’(x)

\[\frac{d}{dx}\][f(g(x))] =  f’(g(x))g’(x)


 The Chain Rule Derivative States that:

The derivative of a composite function can be said as the derivative of the outer function which we multiply by the derivative of the inner function.


Chain Rule Differentiation:

Here are the two functions f(x) and g(x), the chain rule formula is,

( \f∘g )( x ) equals f ′ ( g( x ) )·g′( x )

Let's work some chain rule examples to understand the chain rule calculus in a better rule.

To work these examples it requires the use of different differentiation rules.

Steps to be Followed While Using Chain Rule Formula –

Step 1:

You need to obtain f′(g(x)) by differentiating the outer function and keeping the inner function constant.

Step 2:

Now you need to compute the function g ′ (x), by differentiating the inner function.

Step 3:

Now you just need to express the final answer you have got in the simplified form.

 

NOTE: Here the terms f’(x) and g’(x) represent the differentiation of the functions f(x) and g(x) respectively. Let’s solve chain rule problems.

Questions to be Solved -

Example 1. (5x + 3)2

Step 1:  You need to identify the inner function and then rewrite the outer function replacing the inner function by u.

Let g = 5x + 3 which is the Inner Function

We can now write,

u = 5x + 3      We will set Inner Function to the variable u

f = u2                      This is known as the Outer Function.

Step 2: In the second step, take the derivative of both functions.

The derivative of f = u2

d/dx (u2)                         This is the Original Function.

2u                              This is the power & Constant

The derivative of the function namely g = x + 3

d /dx (5x+3)                        Original function

d /dx ( 5x)+ d/ dx 3                    Use the Sum Rule

5 d/dx( x+3)       We pull out the Constant Multiple

5x0 + 0                                             Power & Constant

We get 5 as the final answer.

Step 3: In the step 3, you need to substitute the derivatives and the original expression for the variable u into the Chain Rule and then you need to simplify.

( f∘g )( x )equals  f ′ ( g( x ) )·g′( x )

2u(5)                               Applying the Chain Rule

2(5x + 3)(5)                     Substitute the value of u

50x + 30                      After simplifying we get this.

ALTERNATIVE WAY!

If the expression is simplified first, then the chain rule is not needed.

Step 1: Simplify the question.

(5x + 3)2

Can be written as , 

(5x + 3)(5x + 3)

25x2 + 15x + 9+ 15x

25x2 + 9 + 30x

Step 2: Now you need to differentiate without the chain rule.

d /dx ( 25x2 + 9 + 30x)                  Original Function

d/ dx (25x2) +d /dx( 30x)+ d /dx (9)        Apply Sum Rule

25 d/ dx(x2 )+30 d/dx (x)+ d/dx (9)

                                           Putting the Constant aside.

25(2x1) + 30x0           Solving for Power & Constant

50x + 30 Answer.

FAQ (Frequently Asked Questions)

Q1. Why Does Chain Rule Work? How do you Solve the Product Rule?

Ans. This rule is known as the chain rule because we use it to take derivatives of composites of functions by chaining together their derivatives. The chain rule can be said as taking the derivative of the outer function ( which is applied to the inner function) and multiplying it times the derivative of the inner function.

 The product rule generally is used if the two "parts" of the function are being multiplied together, and the chain rule is used if the functions are being composed. For instance, to find the derivative of f(x) = x² sin(x), you use the product rule, and to find the derivative of g(x) = sin(x²) you use the chain rule.

Q2. What is the Difference Between Chain Rule and Power Rule?

Ans. The general power rule is a special case of the chain rule. It is very essential when we find the derivative of a function that is raised to the nth power. The general power rule basically states that any given derivative is n times the function and raised to the (n-1)th power times the derivative of the given function. These are two really useful rules for differentiating functions. We use the chain rule when differentiating a 'function of a function', like f(g(x)) in general. We use the product rule when differentiating two functions multiplied together, like f(x)g(x) in general.

Take an example, f(x) = sin(3x). This is an example of what is properly called a 'composite' function; basically a 'function of a function'. The two functions in this example are as follows: function one takes x and multiplies it by 3; function two takes the sine of the answer given by function one. To differentiate these types of functions we generally use the chain rule.