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

• Last Updated : 03 Jan, 2021

### 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:

â‡’

â‡’

â‡’

â‡’

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

### Question 2. Find the derivative of f(x) = x2â€“ 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:

â‡’

â‡’

â‡’

â‡’

â‡’

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:

â‡’

â‡’

â‡’

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:

â‡’

â‡’

â‡’

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:

â‡’

â‡’

â‡’

âˆµ 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

âˆ´

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.

â‡’

â‡’

Using the formula:

âˆ´

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:

â‡’

â‡’

â‡’

âˆ´ Use the formula:  {sandwich theorem}

â‡’

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:

â‡’

â‡’

â‡’  f'(\pi/2)=   {âˆµ

âˆµ 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

âˆ´

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.

â‡’

â‡’

Using the formula:

âˆ´

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:

â‡’

â‡’

â‡’

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:

â‡’

â‡’ f'(\pi/2)= \lim_{h \to 0} \frac {-2\sin(h)} h {âˆµ }

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

âˆ´

Using the formula:

âˆ´

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:

â‡’

â‡’  {âˆµ}

â‡’

â‡’

âˆµ 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:

âˆ´

Hence, derivative of f(x)=

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