Class 11 RD Sharma Solutions- Chapter 30 Derivatives - Exercise 30.1

Question 1. Find the derivative of f(x) = 3x at x = 2

Solution:

Given: f(x)=3x

By using the derivative formula,

 {where h is a small positive number}

Derivative of f(x)=3x at x=2 is given as:

⇒ 

⇒ 

⇒ 

⇒ 

Hence, derivative of f(x)=3x at x=2 is 3

Question 2. Find the derivative of f(x) = x²– 2 at x = 10

Solution:

Given: f(x)= x²-2 

By using the derivative formula,

 {where h is a small positive number}

Derivative of f(x)=x²-2 at x=10 is given as:

⇒

⇒

⇒

⇒

⇒

Hence, derivative of f(x)=x²-2 at x=10 is 20

Question 3. Find the derivative of f(x) = 99x at x = 100

Solution:

Given: f(x)= 99x

By using the derivative formula,

 {where h is a small positive number}

Derivative of f(x)=99x at x=100 is given as:

⇒ 

⇒ 

⇒ 

Hence, derivative of f(x)=99x at x=100 is 99

Question 4. Find the derivative of f(x) = x at x = 1

Solution:

Given: f(x)=x

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)=x at x=1 is given as:

⇒ 

⇒ 

⇒

Hence, derivative of f(x)=x at x=1 is 1

Question 5. Find the derivative of f(x) =  at x = 0

Solution:

Given: f(x)=

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)= at x=0 is given as:

⇒ 

⇒ 

⇒ 

∵ we can not find the limit of the above function f(x)= by direct substitution as it gives 0/0 form (indeterminate form)

So we will simplify it to find the limit.

As we know that 

∴ 

Divide the numerator and denominator by 2 to get the form  for applying sandwich theorem and multiplying h in numerator and denominator to get the required form.

⇒ 

⇒ 

Using the formula: 

∴ 

Hence, derivative of f(x)= at x=0 is 0

Question 6. Find the derivative of f(x) =  at x = 0

Solution:

Given: f(x)=

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)= at x=0 is given as:

⇒ 

⇒ 

⇒ 

∴ Use the formula:  {sandwich theorem}

⇒ 

Hence, derivative of f(x)= at x=0 is 1

Question 7(i). Find the derivatives of the following functions at the indicated points :  at 

Solution:

Given: f(x)= 

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)= at  is given as:

⇒

⇒ 

⇒  f'(\pi/2)=   {∵

∵ we can not find the limit of the above function by direct substitution as it gives 0/0 form (indeterminate form)

So we will simplify it to find the limit.

As we know that

∴ 

Divide the numerator and denominator by 2 to get the form (sin x)/x for applying sandwich theorem and multiplying h in numerator and denominator to get the required form.

⇒ 

⇒ 

Using the formula:

∴

Hence, derivative of f(x)=  at  is 0

Question 7(ii). Find the derivatives of the following functions at the indicated points : x at x=1

Solution:

Given: f(x)=x

By using the derivative formula,

 {where h is a small positive number}

Derivative of f(x)=x at x=1 is given as:

⇒ 

⇒ 

⇒ 

Hence, derivative of f(x)=x at x=1 is 1

Question 7(iii). Find the derivatives of the following functions at the indicated points : 2\cos x at 

Solution:

Given: f(x)= 

By using the derivative formula,

 {where h is a small positive number}

Derivative of f(x)=  at  is given as:

⇒ 

⇒ f'(\pi/2)= \lim_{h \to 0} \frac {-2\sin(h)} h {∵ }

∵ we can not find the limit of the above function by direct substitution as it gives 0/0 form (indeterminate form)

∴ 

Using the formula: 

∴ 

Hence, derivative of f(x)= 

Question 7(iv). Find the derivatives of the following functions at the indicated points :  at 

Solution:

Given: f(x)= 

By using the derivative formula,

 {where h is a small positive number}

Derivative of f(x)=  at  is given as:

⇒ 

⇒  {∵}

⇒ 

⇒ 

∵ we can not find the limit of the above function by direct substitution as it gives 0/0 form (indeterminate form)

Using the sandwich theorem  and multiplying 2 in numerator and denominator to apply the formula.

Using the formula: 

∴ 

Hence, derivative of f(x)=