RD Sharma Class 12 Solutions - Differentiation - 1 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


11. Differentiation
Exercise 11.1
1. Question
Differentiate the following functions from first principles :
e
–x
Answer
We have to find the derivative of e
–x
 with the first principle method, so,
f(x) = e
–x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = – e
–x
2. Question
Differentiate the following functions from first principles :
e
3x
Answer
We have to find the derivative of e
3x
 with the first principle method, so,
f(x) = e
3x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 3e
3x
3. Question
Differentiate the following functions from first principles :
Page 2


11. Differentiation
Exercise 11.1
1. Question
Differentiate the following functions from first principles :
e
–x
Answer
We have to find the derivative of e
–x
 with the first principle method, so,
f(x) = e
–x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = – e
–x
2. Question
Differentiate the following functions from first principles :
e
3x
Answer
We have to find the derivative of e
3x
 with the first principle method, so,
f(x) = e
3x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 3e
3x
3. Question
Differentiate the following functions from first principles :
e
ax + b
Answer
We have to find the derivative of e
ax+b
 with the first principle method, so,
f(x) = e
ax+b
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = a e
ax+b
4. Question
Differentiate the following functions from first principles :
e
cos
 
x
Answer
We have to find the derivative of e
cos
 
x
 with the first principle method, so,
f(x) = e
cos
 
x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 
f ‘(x) = 
[By using cos(x+h) = cosx cosh – sinx sinh]
f ‘(x) = 
[By using lim
x?0
  = 1 and
cos 2x = 1–2sin
2
 x]
f ‘(x) = 
Page 3


11. Differentiation
Exercise 11.1
1. Question
Differentiate the following functions from first principles :
e
–x
Answer
We have to find the derivative of e
–x
 with the first principle method, so,
f(x) = e
–x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = – e
–x
2. Question
Differentiate the following functions from first principles :
e
3x
Answer
We have to find the derivative of e
3x
 with the first principle method, so,
f(x) = e
3x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 3e
3x
3. Question
Differentiate the following functions from first principles :
e
ax + b
Answer
We have to find the derivative of e
ax+b
 with the first principle method, so,
f(x) = e
ax+b
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = a e
ax+b
4. Question
Differentiate the following functions from first principles :
e
cos
 
x
Answer
We have to find the derivative of e
cos
 
x
 with the first principle method, so,
f(x) = e
cos
 
x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 
f ‘(x) = 
[By using cos(x+h) = cosx cosh – sinx sinh]
f ‘(x) = 
[By using lim
x?0
  = 1 and
cos 2x = 1–2sin
2
 x]
f ‘(x) = 
f ‘(x) = 
f ‘(x) = –e
cos x
 sin x
5. Question
Differentiate the following functions from first principles :
Answer
We have to find the derivative of e
v2x
 with the first principle method, so,
f(x) = e
v2x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 
[By rationalising]
f ‘(x) = 
f ‘(x) = 
6. Question
Differentiate each of the following functions from the first principal :
log cos x
Answer
We have to find the derivative of log cosx with the first principle method, so,
f(x) = log cos x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
Page 4


11. Differentiation
Exercise 11.1
1. Question
Differentiate the following functions from first principles :
e
–x
Answer
We have to find the derivative of e
–x
 with the first principle method, so,
f(x) = e
–x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = – e
–x
2. Question
Differentiate the following functions from first principles :
e
3x
Answer
We have to find the derivative of e
3x
 with the first principle method, so,
f(x) = e
3x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 3e
3x
3. Question
Differentiate the following functions from first principles :
e
ax + b
Answer
We have to find the derivative of e
ax+b
 with the first principle method, so,
f(x) = e
ax+b
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = a e
ax+b
4. Question
Differentiate the following functions from first principles :
e
cos
 
x
Answer
We have to find the derivative of e
cos
 
x
 with the first principle method, so,
f(x) = e
cos
 
x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 
f ‘(x) = 
[By using cos(x+h) = cosx cosh – sinx sinh]
f ‘(x) = 
[By using lim
x?0
  = 1 and
cos 2x = 1–2sin
2
 x]
f ‘(x) = 
f ‘(x) = 
f ‘(x) = –e
cos x
 sin x
5. Question
Differentiate the following functions from first principles :
Answer
We have to find the derivative of e
v2x
 with the first principle method, so,
f(x) = e
v2x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 
[By rationalising]
f ‘(x) = 
f ‘(x) = 
6. Question
Differentiate each of the following functions from the first principal :
log cos x
Answer
We have to find the derivative of log cosx with the first principle method, so,
f(x) = log cos x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[Adding and subtracting 1]
f ‘(x) = 
[Rationalising]
f ‘(x) = 
[By using lim
x?0
  = 1]
f ‘(x) = 
[cosC – cosD = –2 sin sin ]
f ‘(x) =  [By using lim
x?0
  = 1]
f ‘(x) = 
f ‘(x) = – tan x
7. Question
Differentiate each of the following functions from the first principal :
Answer
We have to find the derivative of  with the first principle method, so,
f(x) = 
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using lim
x?0
  = 1]
f ‘(x) = 
[Rationalizing]
f ‘(x) = 
Page 5


11. Differentiation
Exercise 11.1
1. Question
Differentiate the following functions from first principles :
e
–x
Answer
We have to find the derivative of e
–x
 with the first principle method, so,
f(x) = e
–x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = – e
–x
2. Question
Differentiate the following functions from first principles :
e
3x
Answer
We have to find the derivative of e
3x
 with the first principle method, so,
f(x) = e
3x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 3e
3x
3. Question
Differentiate the following functions from first principles :
e
ax + b
Answer
We have to find the derivative of e
ax+b
 with the first principle method, so,
f(x) = e
ax+b
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = a e
ax+b
4. Question
Differentiate the following functions from first principles :
e
cos
 
x
Answer
We have to find the derivative of e
cos
 
x
 with the first principle method, so,
f(x) = e
cos
 
x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 
f ‘(x) = 
[By using cos(x+h) = cosx cosh – sinx sinh]
f ‘(x) = 
[By using lim
x?0
  = 1 and
cos 2x = 1–2sin
2
 x]
f ‘(x) = 
f ‘(x) = 
f ‘(x) = –e
cos x
 sin x
5. Question
Differentiate the following functions from first principles :
Answer
We have to find the derivative of e
v2x
 with the first principle method, so,
f(x) = e
v2x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using  = 1]
f ‘(x) = 
[By rationalising]
f ‘(x) = 
f ‘(x) = 
6. Question
Differentiate each of the following functions from the first principal :
log cos x
Answer
We have to find the derivative of log cosx with the first principle method, so,
f(x) = log cos x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[Adding and subtracting 1]
f ‘(x) = 
[Rationalising]
f ‘(x) = 
[By using lim
x?0
  = 1]
f ‘(x) = 
[cosC – cosD = –2 sin sin ]
f ‘(x) =  [By using lim
x?0
  = 1]
f ‘(x) = 
f ‘(x) = – tan x
7. Question
Differentiate each of the following functions from the first principal :
Answer
We have to find the derivative of  with the first principle method, so,
f(x) = 
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using lim
x?0
  = 1]
f ‘(x) = 
[Rationalizing]
f ‘(x) = 
f ‘(x) = 
[sinA cosB – cosA sinB = sin(A–B)]
f ‘(x) = 
[By using lim
x?0
  = 1]
f ‘(x) = 
f ‘(x) = 
8. Question
Differentiate each of the following functions from the first principal :
x
2
 e
x
Answer
We have to find the derivative of x
2
e
x
 with the first principle method, so,
f(x) = x
2
e
x
by using the first principle formula, we get,
f ‘(x) = 
f ‘(x) = 
f ‘(x) = 
[By using (a+b)
2
 = a
2
+b
2
+2ab]
f ‘(x) = 
f ‘(x) = 
[By using lim
x?0
  = 1]
f ‘(x) = x
2
e
x
 + lim
h?0
 e
(x+h)
 [h+2x]
f ‘(x) = x
2
e
x
 + 2x e
x
9. Question
Differentiate each of the following functions from the first principal :
log cosec x
Answer
We have to find the derivative of log cosec x with the first principle method, so,
f(x) = log cosecx
by using the first principle formula, we get,
f ‘(x) =