Differentiation of Implicit Functions MCQ Quiz - Objective Question with Answer for Differentiation of Implicit Functions - Download Free PDF

Calculation:

Given,

and .

Differentiate both sides with respect to implicitly:

Left side:

Right side (product rule):

Rearrange to collect terms and use

.

After cancellation of the common factor , you obtain:

∴ 

Hence, the correct answer is Option 1. 

Calculation:

Given,

Differentiate implicitly w.r.t. :

Rearrange to collect :

Use to simplify:

∴ , independent of and .

Hence, the correct answer is Option 4.

Calculation:

Given,

and .

Differentiate both sides with respect to implicitly:

Left side:

Right side (product rule):

Rearrange to collect terms and use

.

After cancellation of the common factor , you obtain:

∴ 

Hence, the correct answer is Option 1. 

Calculation:

Given,

Differentiate implicitly w.r.t. :

Rearrange to collect :

Use to simplify:

∴ , independent of and .

Hence, the correct answer is Option 4.

Concept:

Calculation:

Given 2x + 2y = 2x+y

We know that 2^a+b = 2^a⋅ 2^b

⇒ 2^x + 2^y = 2^x ⋅ 2^y

⇒ 

⇒ 2^-y + 2^-x = 1  ..(1)

Differentiating the above equation with respect to x:

⇒ (-2^{- y} - 2^{- x}) log 2 = 0

⇒ 

⇒ 

⇒  = - 2^y + 1

⇒  = 1 - 2^y

The required value of   is 1 - 2^y .  

Calculation:

xe = e 

Taking log on both sides, we get

⇒ log xe = log e 

⇒ e log x = (x² + y²) log e             

[∵ log mⁿ = n log m]

⇒ e log x = x2 + y2                       

[∵ log e = 1]

Differentiating w.r.t x, we get

⇒ e() = 

⇒ 

∴ 

Concept:

Calculus:

• .
•  

Chain Rule of Derivatives:

• .

Calculation:

Given expression is:

3x + 3y = 3x + y.

Differentiating w.r.t. x and using the chain rule of derivatives, we get:

⇒ 3x (log 3) + 3y (log 3)  = 3x + y (log 3) (1 + )

⇒ 3x + 3y  = 3x + y + 3x + y 

⇒ 

Calculation:

Here, 

Differentiating w.r.t. x, we get

Hence, option (3) is correct.

Calculation:

y + sin-1 (1 - x2) = ex 

y = e^x - sin^-1 (1 - x²)

Differentiating w.r.t x, we get

Concept:

Calculation:

Given: 

Differentiating with respect to x, we get

Concept:

Calculation:

Given: y2 = 4ax

Differentiating with respect to x, we get

Given

y = 3e^2x + 2e^3x

Formula used

d(xⁿ)/dx = nx^n-1

Solution

⇒dy/dx = 3e^2x(2) + 2e^3x(3)

⇒dy/dx = 6(e^2x + e^3x)

⇒d²y/dx² = 6(2e^2x+3e^3x)

As asked in the question,

⇒   - 5  + 6y =

⇒ 12e^2x + 18e^3x − 30e^2x − 30e^3x + 18e^2x + 12e^3x

⇒ 0.

The correct option is 4.

Concept:

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

Calculation:

Given y2 + x2 + 3x + 5 = 0

Differentiating with respect to x, we get

2y  + 2x +3(1) + 0 = 0

2y + 2x + 3 = 0 

2y  = -(2x + 3)

Now at (0, -3)

 = 0.5

Calculation:

Given function is y² = 2x;

Differentiating with respect to x on both sides,

2yy' = 2x ⇒ yy’ = 1   - (1)

⇒ y’ = 1/y;

Differentiating (1) again with respect to x on both sides,

⇒ y . y” + y’. y’ = 0   - (2)

Concept:

cos2x = 2cos²x - 1

sin2x = 2sin x cos x

Calculation:

Here, y = cos2 x2

Let, x² = t 

Differentiating with respect to x, we get

⇒2xdx = dt 

⇒ dt/dx = 2x ....(1)

y = cos²t

=

= - sin2x² × 2x

= -4x cos x2 sin x2

Hence, option (2) is correct.