The chain rule is an important topic of Quantitative Aptitude that needs to be practiced well for competitive exams. The following article includes the concepts, steps, and formulas that are used to solve the chain rule questions.
What is Chain Rule?
This chain rule is also referred to as the outside-inside rule, the composite function rule, or the function of a function rule. It is only used to determine the derivatives of composite functions.
The chain rule is used to calculate the derivative of composite functions like- (3x2 + 2x- 6)5, sin(x2 +5x), e2x, (4x2 +5x)(2x+6), etc. The Chain Rule is thought to have first originated from the German mathematician Gottfried W. Leibniz. Although the memoir it was first found in contained various mistakes, it is apparent that he used the chain rule in order to differentiate a polynomial inside of a square root.
Tips and Tricks to Solve Chain Rule Questions
Step 1: Find The Chain Rule: It must be a composite function, which should contain a nested function.
Step 2: Find out the inner function and the outer function.
Step 3: First of all, find the derivative of the outer function, after that inner function.
Step 4: Find the product of results obtained from step4 and step5.
Step 5: At last, Simplify the derivative of the chain rule.
Chain Rule Formulas
There are two forms of chain rule formula as shown below.
Chain Rule Formula 1
d/dx ( f(g(x) ) = f’ (g(x)) · g’ (x)
Example: To find the derivative of d/dx .(log5x),
f(g(x))=log5x
where, f(x) = log(x) and, g(x) = 5x
By chain rule,
d/dx .(log5x) = 5/5x = 1/x.
Chain Rule Formula 2
dy/dx = dy/du · du/dx
Example: To find the derivative of d/dx .(log5x),
Suppose, y=log5x, and u=5x.
By the chain rule formula,
d/dx. (log5x) = d/du .(log5x) . du/dx
=1/5x .5
d/dx. (log5x) =1/x.
Double Chain Rule
Functions that depend on multiple variables may be nested one on top of the other. To obtain the total derivative, the series of smaller derivatives are multiplied together. Let’s say there are 3 functions: u, v, and w. A function is a combination of u, v, and w. Here, the chain rule is broadened. The chain rule is used twice when a function is made up of three different functions.
If f = (u o v) o w = df/dx = df/du. du/dv. dv/dw. dw/dx
Example1: y = (1+tan3x)2
y’ = 6(1+tan3x). sec23x
Example 2: y = (2x-5)2
y’ = 4(2x-5)
Chain Rule Questions and Answers
Que 1. Find the derivative of the function sin (ax+b)
Solution:
Given function is:
f(x) = sin(ax+b) [it is a composite function]
Differentiate with respect to x,
d/dx (f(x)) = d/dx(sin(ax+b))
By the chain rule formula,
dy/dx = dy/du · du/dx
f''(x)= d sin(ax+b)/d(ax+b) × d(ax+b)/ dx
f'(x)= cos(ax+b) × [ d(ax)/dx + d(b)/dx]
= cos(ax+b) × [a × 1 + 0 ]
=cos(ax+b) × a
f'(x) =a cos(ax+b)
Que 2. Find the derivative of the function , f(x)= (3x+4)2
Solution:
Given function is:
f(x)=(3x+4)2.
Differentiate with respect to x,
d/dx (f(x)) = d/dx (3x+4)2
By the chain rule formula,
dy/dx = dy/du · du/dx
f''(x)=d(3x+4)2 / d(3x+4) × d(3x+4)/ dx
f'(x)= 2 (3x+4) × [d(3x)/dx + d(4)/dx]
f'(x) = 2(3x+4) × [3 × 1 + 0]
f'(x) = 2(3x+4) × 3
f'(x)= 6(3x+4)
Que 3. Find the derivative of the function f(x) = log(2x2 + 5)
Solution:
Given function is :
f(x) = log(2x2+ 5)
The given function is composite function so,
we are using chain rule to solve the problem.
By chain rule formula,
dy/dx = dy/du . du/dx
f '(x) = d(log(2x2 +5)) / d(2x2 +5) . d(2x2 +5) / dx
= 1/(2x2+5) . 4x
f '(x) = 4x / (2x2+5)
Que 4. Find the derivative of the function f(x) = √(6x + 5)
Solution:
Given function is:
f(x) = √(6x + 5)
The given function is composite function so,
we are using chain rule to solve the problem.
By chain rule formula,
dy/dx = dy/du . du/dx
f '(x) =d(√(6x + 5)) / d(6x + 5) . d(6x + 5) /dx
f '(x)= 1/2 (√(6x + 5)) . 6
f '(x) =3 / √(6x + 5)
Que 5. Find dy/dx if y = 4x^3 + 2x^2 + 5x – 3?
Solution:
To find the derivative of y with respect to x,
we need to take the derivative of each
term separately. Using the power rule, we get:
dy/dx = 12x^2 + 4x + 5
Therefore, the derivative of y with respect to x is 12x^2 + 4x + 5.
FAQs on Chain Rule
Que 1. What is Chain Rule?
Answer: The chain rule is used to calculate the derivative of composite functions like- (3×2 + 2x- 6)5, sin(x2 +5x), e2x, (4×2 +5x)(2x+6) etc. This chain rule is also referred to as the outside-inside rule, the composite function rule, or the function of a function rule. It is only used to determine the derivatives of composite functions.
Que 2. What is Chain Rule Formula?
Answer: The chain rule is used to calculate the derivative of composite functions (i.e when one function is inside the other). the chain rule formula is used in two ways:
d / dx (f(g(x))) = f' (g(x))·g' (x),
dy/dx = dy/du . du/dx
Que 3. When to Use Chain Rule Formula?
Answer: When we have to find the derivative of the functions, which have a function inside functions (Composite function), we use the Chain rule formula, Because we don’t have a direct formula for composite functions, we have to use the Chain rule formula in that case.
The chain rule formula is : d / dx (f(g(x))) = f’ (g(x))·g’ (x),
Que 4. What are the applications of the Chain Rule Formula?
Answer: The chain rule is used as the main tool to solve Implicit Differentiation, Logarithmic Differentiation, Inverse Functions Differentiation, and Related Rates(rate of change of distance, rate of change of the average molecular speed, etc.).
Que 5. What is the difference between the chain rule formula and the product rule?
Answer: The chain rule formula is to find the derivative of a composite function(a function that is combined with other functions). ex- (2x2 + 5x – 6) while the product rule formula is used to find the derivative of the product of two or more functions. ex- (x +3). (x – 6).
The chain rule formula is d / dx (f(g(x))) = f’ (g(x))·g’ (x), while, The product rule formula is : d / dx (u.v) = u. dv/dx + v. du/dx
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