Chain Rule of Differentiation: Steps and Solved Examples

The Chain Rule formula computes the derivative of the composite of two or more functions. The chain rule for the differentiation of composite functions is defined. For example, if f and g are functions, then the chain rule expresses their composition's derivative.

d/dx [f(g(x))] = f'(g(x)) g'(x)

Key Terms: Derivative, Function, Constant, Trigonometric, Calculus, Limits, Continuity, Logarithmic, Exponential

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What is Differentiation?

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Rates of change can be found by using differentiation. Differentiation lets us find, for example, the rate of change of velocity in relation to time (which gives the acceleration).

• The idea of differentiation also lets us find the rate of change of the variable x in relation to the variable y. 
• This rate is called the gradient of the curve when it is shown on a graph of y against x. 
• The chain rule can be written as f o g, where f and g are two functions. 
• Now to find the difference between functions that are known to be composite there are a few steps:

1. For instance, let's look at the function (x + 3)². The inner function, g, is equal to (x + 3), and if x + 3 = u, then the outer function, f, can be written as u² times the inner function.
2. This rule is also called the chain rule because it is used to find the derivatives of functions that are made up of other functions. This is done by chaining the derivatives of the functions together.
3. The chain rule can be thought of as multiplying the derivative of the outer function (which is used to calculate the inner function) by the derivative of the inner function.

d/dx[f(x)ⁿ]=n(f(x))ⁿ−1.f’(x)

d/dx[f(x)ⁿ]=f’(g(x))g’(x)

Read More: Non-Parametric Test

Discover about the Chapter video:

Continuity and Differentiability Detailed Video Explanation:

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What is Chain Rule Formula?

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When one multiplies the derivative of the outer function by the derivative of the inner function, we get the derivative of the composite function.

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

(fog )( x ) = f ′ ( g( x ) )·g′( x )

Read More: Infinity

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Steps to Solve Chain Rule Formula

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Here are the terms f’(x) and g’(x) represent the differentiation of the functions f(x) and g(x), respectively.

1. In order to produce f'(g(x)), one will need to differentiate the outer function while maintaining the consistency of the inner function.
2. One must now differentiate the inner function in order to compute the function g'(x), which stands for "x."
3. Now all that is required of you is to express the final answer you have gotten in a form that is simple.

Read More: Differentiation Rules

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Common Mistakes Made in Chain Rule

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Here are some of the most common mistakes students make when using the chain rule formula.

• One often misidentifies a function as composite since the chain rule is the sole way to separate a composite function. 
• Without it, the function will be erroneously derived.
• After realizing the function is composite, one may misidentify inner and outer functions, resulting in an incorrect derivative.
• One commonly forgets to multiply by the derivative of the inner function while applying the chain rule. 
• One usually differentiates the outer function without deriving the inner function, which is incorrect.

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Solved Examples

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Example 1: Compute the derivative of the function y = arcsin(2x + 1), i.e., y = sin^-1(2x + 1).

Solution: Given,

y = arcsin(2x + 1)

= sin^-1(2x + 1)

Let y = sin-1(u) and u = 2x + 1.

dy/du = (d/du) sin-1(u)

= 1/√(1 – u²)

du/dx = (d/dx)(2x + 1)

= 2

Using the chain rule;

dy/dx = (dy/du)(du/dx)

= [1/√(1 – u²)] (2)

= 2/√[1 – (2x + 1)2]

= 2/√[1 – (4x² + 1 + 4x)]

= 2/√(1 – 4x² – 1 – 4x)

= 2/√( – 4x² – 4x)

= 2/2√[-x(x + 1)]

= 1/√[-x(x + 1)]

Hence, dy/dx = 1/√[-x(x + 1)].

Example 2: Find dy/dx if y = 2 sin(3 cos 4x).

Solution: Given,

y = 2 sin(3 cos 4x)

Let u = 3 cos 4x and y = 2 sin u

du/dx = 3(-sin 4x)(d/dx)4x

= -12 sin 4x

And dy/du = (d/du) (2 sin u) = 2 cos u

Using the chain rule;

dy/dx = (dy/du) (du/dx)

= (2 cos u) (-12 sin 4x)

= -24 sin 4x cos(3 cos 4x)

Therefore, dy/dx = -24 sin 4x cos(3 cos 4x).

Read More: Differential Equations

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Things to Remember

• The chain rule differentiates composite functions. Consequently, h'(x) = f'(g (x)). g’(x). 
• Leibniz's chain rule is dy/dx=(dy/du)*(du/dx).
• Formulas can be derived using the chain rule and other principles.
• Combining the chain and power rules creates a new rule. 
• h'(x) is the product of g'(x) and n(g(x)) to the power of n. (n - 1).
• In order to produce f'(g(x)), one will need to differentiate the outer function while maintaining the consistency of the inner function.
• If the chain rule is applied to the composition of three functions, the rule h(x) = f(g(k(x))) becomes h'(x) = f'(g(k(x))). g’(k(x)) . k’(x)

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Previous Year Questions

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