Second Order Derivatives MCQ Quiz - Objective Question with Answer for Second Order Derivatives - Download Free PDF

Calculation:

⇒ 

⇒ y' - xy' - y = -32x³¹

⇒ y" - xy" - y' - y' = -(32)(31)x³⁰

at x = – 1 ⇒ y' - y" = 496

Hence, the correct answer is Option 3. 

Calculation: 

Applying C₃ → C₃ -C₁

Hence, the correct answer is Option 1.

Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

• Chain Rule: 
• Product Rule: 

Formulas:

Calculation:

= 

= 

= 

Calculation:

 y = 100e2x - 200e-2x

Differentiate y with respect to x:

⇒

Differentiate again with respect to x:

⇒

Factor out 4:

⇒

⇒

Comparing with , we get a = 4.

Hence option 1 is correct

Formula Used:

Chain rule for differentiation

Calculation:

Given:

Differentiating both sides w.r.t. x, we get

⇒

Differentiating again w.r.t. x, we get

⇒

Now, substituting the values of y and y'' in the expression , we get

⇒

⇒

⇒

∴

Hence option 1 is correct

Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

• Product Rule: 
• Division rule: 

Formulas:

Calculation:

= 

= 

= 

⇒ 

⇒

= 

Concept:

Calculation:

Given: y = p cos 2x + q sin 2x                    .... (1)

Differentiating with respect to x, we get

As we know that, 

Again, differentiating with respect to x, we get

From equation (1), we get

Concept:

Chain Rule of Derivatives:

• .
• .

Product Rule of Derivatives:

• .

Calculation:

We have x =  and y = .

We observe that x + y =  = 1.

Differentiating w.r.t. x, we get:

1 +  = 0

⇒  = -1

Differentiating again w.r.t. x, we get:

.

CONCEPT:

0\)

CALCULATION:

Given: y = sin (log x)

First let's find out y₁

As we know that,  and  0\)

⇒ 

Now, again by differentiating the above equation with respect to x we get,

As we know that, 

⇒ 

Now, x²y₂ + xy₁ = -y

Hence, correct option is 2.

Concept:

Calculation:

To Find: 

= 20 × 19 × x¹⁸

= 380x18 

Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

• Chain Rule: 
• Product Rule: 

Formulas:

Calculation:

= 

= 

= 

= 

Concept:

Chain Rule of Derivatives:

• .
• .

Derivatives of Trigonometric Functions:

Calculation:

Using the chain rule of derivatives, we get:

 = 2 sec² 2x

Again differentiation in respect to x, we get

= 

=2 (2 sec 2x)(tan 2x sec 2x)(2)

= 8 tan 2x sec² 2x

Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

• Chain Rule: 
• Product Rule: 

Formulas:

Calculation:

We have to find the value of 

Apply chain rule, we get

= 2sec x . sec x tan x

= 2sec² x tan x

Concept:

Calculation:

To Find: 

= 10 × 9 × x8

= 90(x8)

CONCEPT:

CALCULATION:

Given: y = 5 cos x - 3 sin x

First let's find out dy/dx

As we know that,  and 

⇒ dy/dx = - 5 sin x - 3 cos x

Now, again by differentiating the above equation with respect to x we get,

Hence, correct option is 4.