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.