# 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.