Differentiation by taking log MCQ Quiz in తెలుగు - Objective Question with Answer for Differentiation by taking log - ముఫ్త్ [PDF] డౌన్‌లోడ్ కరెన్

Concept Used:

Power Rule: 

Product Rule: 

Chain Rule: 

Logarithmic Differentiation

Calculation

Given:

⇒

Let . ⇒ .

⇒ .

⇒ .

Using product rule:

⇒

At :

⇒

⇒

⇒

⇒

⇒

∴ The value is .

Hence option 3 is correct.

Given,

Taking on both sides, we get

If

Taking log on both sides,

On differentiating w.r.t. x, we get

Concept:

Product rule: (f.g)'(x) = f '(x).g(x) + f(x).g'(x)

Formula used :

• (ln x)' = 

Calculation:

Given,  y = x,

Taking log on both sides,

 ln y =  ln x

Differentiating both sides,

⇒ 

⇒ 

⇒ 

⇒ 

∴ The correct option is (5).

Concept:

• xn = nxn-1
• sin x = cos x
• cos x = -sin x
• ex = ex
• ln x = 
• (ax + b) = a
• tan x = sec2 x
• f(x)g(x) = f'(x)g(x) + f(x)g'(x)
•  sin-1 x = 
•  tan-1 x = 

Calculation:

Given: f(x) = 

Taking log both sides, we get

⇒ log f(x) = log 

⇒ log f(x) = sin x [log(log x)]               (∵ log mn = n log m)

Let f(x) = y

Differentiating with respect to x, we get

 = 

 = y × 

 = 

Calculation:

x^y = e^{x - y} 

Taking log on both sides, we get 

⇒ xy = log ex - y 

⇒ y log x = (x - y) log_e e 

⇒ y log x = x - y             [∵ loge e = 1]

⇒ ( 1 + log x)y = x  

⇒ y = 

Differentiating both sides , we get

 =    

=  

= 

Concept:

Parametric Form:

If f(x) and g(x) are the functions in x, then 

 =  

Calculation:

Let z = x^{-ln x} 

Taking log both sides, we get

ln z = ln x-ln x

ln z = - (ln x)²                       (∵ ln mⁿ = n ln m)

Differentiating with respect to x

 = -2 ln x

Also y = 

Taking log both sides, we get

ln y = ln         

ln y = x² ln e               (∵ ln mn = n ln m)

ln y = x²                      (∵ ln e = 1)

Differentiating with respect to x

The required result is

 = 

= 

= 

Concept:

log₁₀(e) = 1 and

(log₁₀(x))^y = ylog(x)

Calculation:

Given:

For infinite terms, the above equation can be written as:

Taking log both sides

log(x) = (y + x)loge

log(x) = y + x

Differentiating both sides:

Hence option (2) is the solution.