RD Sharma Class 11 Solutions Chapter - Derivatives | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


30. Derivatives
Exercise 30.1
1. Question
Find the derivative of f(x) = 3x at x = 2
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very 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
2. Question
Find the derivative of f(x) = x
2
 – 2 at x = 10
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x
2
 – 2 at x = 10 is given as –
? f’(10) = 0 + 20 = 20
Hence,
Derivative of f(x) = x
2
 – 2 at x = 10 is 20
3. Question
Find the derivative of f(x) = 99x at x = 100.
Answer
Page 2


30. Derivatives
Exercise 30.1
1. Question
Find the derivative of f(x) = 3x at x = 2
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very 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
2. Question
Find the derivative of f(x) = x
2
 – 2 at x = 10
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x
2
 – 2 at x = 10 is given as –
? f’(10) = 0 + 20 = 20
Hence,
Derivative of f(x) = x
2
 – 2 at x = 10 is 20
3. Question
Find the derivative of f(x) = 99x at x = 100.
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of 99x at x = 100 is given as –
Hence,
Derivative of f(x) = 99x at x = 100 is 99
4. Question
Find the derivative of f(x) = x at x = 1
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x at x = 1 is given as –
Hence,
Derivative of f(x) = x at x = 1 is 1
5. Question
Find the derivative of f(x) = cos x at x = 0
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of cos x at x = 0 is given as –
Page 3


30. Derivatives
Exercise 30.1
1. Question
Find the derivative of f(x) = 3x at x = 2
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very 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
2. Question
Find the derivative of f(x) = x
2
 – 2 at x = 10
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x
2
 – 2 at x = 10 is given as –
? f’(10) = 0 + 20 = 20
Hence,
Derivative of f(x) = x
2
 – 2 at x = 10 is 20
3. Question
Find the derivative of f(x) = 99x at x = 100.
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of 99x at x = 100 is given as –
Hence,
Derivative of f(x) = 99x at x = 100 is 99
4. Question
Find the derivative of f(x) = x at x = 1
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x at x = 1 is given as –
Hence,
Derivative of f(x) = x at x = 1 is 1
5. Question
Find the derivative of f(x) = cos x at x = 0
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of cos x at x = 0 is given as –
? we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
So we need to do few simplifications to evaluate the limit.
As we know that 1 – cos x = 2 sin
2
(x/2)
Dividing the numerator and denominator by 2 to get the form (sin x)/x to apply sandwich theorem, also
multiplying h in numerator and denominator to get the required form.
Using algebra of limits we have –
Use the formula: 
? f’(0) = – 1×0 = 0
Hence,
Derivative of f(x) = cos x at x = 0 is 0
6. Question
Find the derivative of f(x) = tan x at x = 0
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of cos x at x = 0 is given as –
? we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
? Use the formula:  {sandwich theorem}
? f’(0) = 1
Hence,
Derivative of f(x) = tan x at x = 0 is 1
7 A. Question
Page 4


30. Derivatives
Exercise 30.1
1. Question
Find the derivative of f(x) = 3x at x = 2
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very 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
2. Question
Find the derivative of f(x) = x
2
 – 2 at x = 10
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x
2
 – 2 at x = 10 is given as –
? f’(10) = 0 + 20 = 20
Hence,
Derivative of f(x) = x
2
 – 2 at x = 10 is 20
3. Question
Find the derivative of f(x) = 99x at x = 100.
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of 99x at x = 100 is given as –
Hence,
Derivative of f(x) = 99x at x = 100 is 99
4. Question
Find the derivative of f(x) = x at x = 1
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x at x = 1 is given as –
Hence,
Derivative of f(x) = x at x = 1 is 1
5. Question
Find the derivative of f(x) = cos x at x = 0
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of cos x at x = 0 is given as –
? we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
So we need to do few simplifications to evaluate the limit.
As we know that 1 – cos x = 2 sin
2
(x/2)
Dividing the numerator and denominator by 2 to get the form (sin x)/x to apply sandwich theorem, also
multiplying h in numerator and denominator to get the required form.
Using algebra of limits we have –
Use the formula: 
? f’(0) = – 1×0 = 0
Hence,
Derivative of f(x) = cos x at x = 0 is 0
6. Question
Find the derivative of f(x) = tan x at x = 0
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of cos x at x = 0 is given as –
? we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
? Use the formula:  {sandwich theorem}
? f’(0) = 1
Hence,
Derivative of f(x) = tan x at x = 0 is 1
7 A. Question
Find the derivatives of the following functions at the indicated points :
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of sin x at x = p/2 is given as –
 {? sin (p/2 + x) = cos x }
? we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
So we need to do few simplifications to evaluate the limit.
As we know that 1 – cos x = 2 sin
2
(x/2)
Dividing the numerator and denominator by 2 to get the form (sin x)/x to apply sandwich theorem, also
multiplying h in numerator and denominator to get the required form.
Using algebra of limits we have –
Use the formula: 
? f’(p/2) = – 1×0 = 0
Hence,
Derivative of f(x) = sin x at x = p/2 is 0
7 B. Question
Find the derivatives of the following functions at the indicated points :
x at x = 1
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
Page 5


30. Derivatives
Exercise 30.1
1. Question
Find the derivative of f(x) = 3x at x = 2
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very 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
2. Question
Find the derivative of f(x) = x
2
 – 2 at x = 10
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x
2
 – 2 at x = 10 is given as –
? f’(10) = 0 + 20 = 20
Hence,
Derivative of f(x) = x
2
 – 2 at x = 10 is 20
3. Question
Find the derivative of f(x) = 99x at x = 100.
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of 99x at x = 100 is given as –
Hence,
Derivative of f(x) = 99x at x = 100 is 99
4. Question
Find the derivative of f(x) = x at x = 1
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x at x = 1 is given as –
Hence,
Derivative of f(x) = x at x = 1 is 1
5. Question
Find the derivative of f(x) = cos x at x = 0
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of cos x at x = 0 is given as –
? we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
So we need to do few simplifications to evaluate the limit.
As we know that 1 – cos x = 2 sin
2
(x/2)
Dividing the numerator and denominator by 2 to get the form (sin x)/x to apply sandwich theorem, also
multiplying h in numerator and denominator to get the required form.
Using algebra of limits we have –
Use the formula: 
? f’(0) = – 1×0 = 0
Hence,
Derivative of f(x) = cos x at x = 0 is 0
6. Question
Find the derivative of f(x) = tan x at x = 0
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of cos x at x = 0 is given as –
? we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
? Use the formula:  {sandwich theorem}
? f’(0) = 1
Hence,
Derivative of f(x) = tan x at x = 0 is 1
7 A. Question
Find the derivatives of the following functions at the indicated points :
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of sin x at x = p/2 is given as –
 {? sin (p/2 + x) = cos x }
? we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
So we need to do few simplifications to evaluate the limit.
As we know that 1 – cos x = 2 sin
2
(x/2)
Dividing the numerator and denominator by 2 to get the form (sin x)/x to apply sandwich theorem, also
multiplying h in numerator and denominator to get the required form.
Using algebra of limits we have –
Use the formula: 
? f’(p/2) = – 1×0 = 0
Hence,
Derivative of f(x) = sin x at x = p/2 is 0
7 B. Question
Find the derivatives of the following functions at the indicated points :
x at x = 1
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of x at x = 1 is given as –
Hence,
Derivative of f(x) = x at x = 1 is 1
7 C. Question
Find the derivatives of the following functions at the indicated points :
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}
? derivative of 2cos x at x = p/2 is given as –
 {? cos (p/2 + x) = – sin x }
? we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
? f’(p/2) = 
Use the formula: 
? f’(p/2) = – 2×1 = – 2
Hence,
Derivative of f(x) = 2cos x at x = p/2 is – 2
7 D. Question
Find the derivatives of the following functions at the indicated points :
Answer
Derivative of a function f(x) at any real number a is given by –
 {where h is a very small positive number}