Chain Rule MCQ Quiz - Objective Question with Answer for Chain Rule - Download Free PDF

Calculation:

Let y = e^{cos x}.

Using the chain rule:

  y' =  

⇒ 

⇒

Conclusion:

∴ The derivative of e^{cos x} with respect to x is:  -e^{cos x} × sin x

Correct Answer: Option 1

Explanation:

Given:

f(0) = -1 and f'(0) = 1 Let h(x) = f(2f(x) +2) and g(x) = (h(x))2 

⇒ g(x) = (h(x))²

⇒ g'(x) = 2(h(x)).h'(x)

⇒ g'(0) = 2(h(0)).h'(0)

= 2f(2f (0) + 2).2

= 2f(0).2 = (–2).2 = –4

∴ Option (a) is correct

Explanation:

Let f: (–1, 1)→ R be a differentiable function with f(0) = –1 and f'(0) = 1

⇒ h(x) = f(2f(x) + 2)

⇒ h'(x) = f'(2f(x) + 2)2f'(x)

⇒ h'(0) = f'(2f(0) + 2).2f'(0)

= f'(–2 + 2).2(1)

= f'(0).2 = (1).2 = 2

∴ Option (d) is correct.

Given:

Differentiate sin(x² + 9) with respect to x.

Formula used:

Chain Rule: If y = sin(u) and u = x² + 9, then dy/dx = cos(u) × du/dx

Calculation:

Let y = sin(x² + 9)

⇒ dy/dx = cos(x² + 9) × (d/dx)(x² + 9)

⇒ dy/dx = cos(x² + 9) × 2x

⇒ dy/dx = 2x cos(x² + 9)

∴ The correct answer is option (4).

Given:

Differentiate sin(x² + 9) with respect to x.

Formula used:

Chain Rule: If y = sin(u) and u = x² + 9, then dy/dx = cos(u) × du/dx

Calculation:

Let y = sin(x² + 9)

⇒ dy/dx = cos(x² + 9) × (d/dx)(x² + 9)

⇒ dy/dx = cos(x² + 9) × 2x

⇒ dy/dx = 2x cos(x² + 9)

∴ The correct answer is option (4).

Concept:

Chain Rule of Derivatives: 

.

 = e^x.

Calculation:

It is given that y = .

∴ y = 

Differentiating both sides with respect to x and using the chain rule, we get:

⇒ 

⇒ 

⇒ 

⇒ 

⇒ .

f'(x) = g(x) and g’(x) = f(x²), then f’’(x²)

given f’(x) = g(x)

Differentiating w.r.t x we get

f’’(x) = g’(x)

f’’(x) = f(x²)

multiply the function with x²

f’’(x²) = f(x⁴)

Concept:

Differentiation Formulas

 = cos x 

 = - sin x

Trigonometry Formula

Calculation:

Given:y = sin (log cos x)

Differentiation with respect to x

 = 

= cos (log (cos x)). 

= cos (log (cos x)).  

= cos (log (cos x)).  .(- sin x)

= - cos (log (cos x)). 

= - cos (log (cos x)).tan x​​

 = - cos (log (cos x)).tan x

Concept:

Calculation:

Given:

Now,

Hence, option (3) is correct.

Concept:​

 sin^-1 x = 

 tan^-1 x = 

Chain Rule (Differentiation by substitution): If y is a function of u and u is a function of x

Calculation:

Let u = 1 - x²

 = - 2x

y = sin^-1(1 - x²) = sin^-1 u

=             

=  sin^-1 u × (-2x)

=  × (-2x)

 = 

= 

= 

Concept:

Given:

u =   And v = 

Calculation:

Let,

x = tan θ 

Then, 

u =  = 

u = 2θ 

 2

And,

v =  =  = 

v = 2θ 

 2

After that,

= 1

Concept:

Differentiation Formulas

 = cos x 

​

Calculation:

Given:

=  

=    

=     

=      

=  ​ . 3 .

= ​

Concept:

Derivative of sinx with respect to x is cosx

Chain rule:

Let y = f(v) be a differentiable function of v and v = g(x) be a differentiable function of x then 

Calculation:

Given function is  y = sin2(2x+5)

We differentiate the function with respect to x

⇒  y' = [sin2(2x+5)]' 

As we know that, 

⇒ y' = 2 sin(2x+5) ⋅ [sin(2x+5)]'

⇒  y' = 2 sin(2x+5) ⋅ cos(2x+5) ⋅ (2x+5)'

⇒ y' = 2 sin(2x+5).cos(2x+5).(2)

⇒ y' = 2 sin[2(2x+5)]                      (∴ sin2x = 2sinx.cosx)

⇒ y' = 2 sin(4x+10)

Hence, option 4 is correct.

Calculation:

Given:

y = log[log{log(x)}] 

DIfferentiation with respect to x

⇒      

Given:

 y = 2x + x log x

Concept:

Use formula

Calculation:

 y = 2x + x log x

Differentiate with respect to x

= 2x log 2 + log x + 1

Hence the option (4) is correct.