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.

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.