### Exercise 13.1

### Evaluate the following limits in Exercises 1 to 22.

**Question 1. **

**Solution :**

Given that,

Substituting x = 3, we get

= 3 + 3 = 6

**Question 2. **

**Solution :**

Given that,

Substituting x = π, we get

=(π – 22 / 7)

**Question 3. **

**Solution :**

Given that,

Substituting r = 1, we get

= ** **π(1)^{2} = π

**Question 4. **

**Solution :**

given limit,

Substituting x = 4, we get

= [4(4) + 3] / (4 – 2) =

= (16 + 3) / 2

= 19 / 2

**Question****5. **

**Solution :**

Given that,

Substituting x = -1, we get

= [(-1)^{10} + (-1)^{5} + 1] / (-1 – 1)

= (1 – 1 + 1) / – 2

= – 1 / 2

**Question****6. **

**Solution :**

Given limit,

Putting x as 0

= [(0 + 1)^{5} – 1] / 0

=0

Since, this limit is undefined

Substitute x + 1 = y, then x = y – 1

=5

**Question 7. **

**Solution :**

**Question 8. **

**Solution :**

**Question 9. **

**Solution :**

Given that,

= [a (0) + b] / c (0) + 1

= b / 1

= b

**Question 10. **

**Solution :**

**Question 11. **

**Solution :**

Given that,

Substituting x = 1

= [a (1)^{2} + b (1) + c] / [c (1)^{2} + b (1) + a]

= (a + b + c) / (a + b + c)

= 1

**Question 12. **

**Solution :**

**Question 13. **

**Solution :**

Given that

**Question 14. **

**Solution :**

**Question 15. **

**Solution :**

**Question 16. **

**Solution :**

**Question 17. **

**Solution :**

**Question****18. **

**Solution :**

**Question****19. **

**Solution :**

**Question****20. **

**Solution :**

**Question****21. **

**Solution :**

**Question****22. **

**Solution :**

= 1 x 2

= 2

**Question****23.****Find and where **

**Solution :**

**Question 24. Find where **

**Solution :**

Given:

.

**Question 25. Evaluate where **

**Solution :**

Given:

.

**Question****26. Find Where **

**Solution :**

Given:

**Question 27. Find where **

**Solution :**

Given:

**Question 28. Suppose and if what are possible values of a and b?**

**Solution :**

Given: and

**Question 29. Let a _{1}, a_{2,}………a_{n }be fixed real numbers and define a function f (x) = (x – a_{1}) (x – a_{2}) ……. (x – a_{n}).What is For some a ≠ a_{1}, a_{2}, ……. a_{n}, compute **

**Solution :**

**Question****30. If for what values of (s) does exists?**

**Solution :**

**Question 31. If the function f(x) satisfies valuate **

**Solution :**

**Question****32. If for what integer m and n does both and exist?**

**Solution :**

Given function is

### Exercise 13.2

### Evaluate the following limits in Exercises 1 to 22.

**Question****1. Find the derivative of x ^{2} – 2 at x = 10.**

**Solution :**

Here f(x) = x^{2} – 2

**Question****2.Find the derivative of 99x at x = 100.**

**Solution :**

Let f (x) = x

Then,

**Question****3. Find the derivative of x at x=1.**

**Solution :**

Let f (x) = 99x,

From first principle

**Question****4. Find the derivatives of the following functions from first principle:**

**(i) x ^{3}−27 **

**(ii) (x−1)(x−2)**

**(iii)1/x ^{2}**

**( iv )x+1/x−1**

**Solution :**

(i)Let f (x) = x^{3} – 27

From first principle

(ii) Let f (x) = (x – 1) (x – 2)

From first principle

(iii)Let f (x) = 1 / x^{2}

From first principle, we get

(iv)Let f (x) = x + 1 / x – 1

From first principle, we get

**Question****5. For the functionprove that f’ (1) =100 f’ (0).**

**Solution :**

**Question****6. Find the derivative of x ^{n} + ax^{n−1 }+ a^{2}x^{n−2} + …+a^{n−1}x + a^{n} for some fixed real number a.**

**Solution :**

**Question****7. For some constants a and b find the derivative of:**

**(i) (x − a) (x − b)**

**(ii) (ax ^{2} + b)^{2}**

**(iii) x – a / x – b**

**Solution :**

(i) Given:(x − a) (x − b)

(ii)(ax^{2} + b)^{2}

(iii) x – a / x – b

**Question****8. Find the derivative offor some constant a.**

**Solution :**

**Question****9. Find the derivative of**

**(i) 2x−3/4**

**(ii) (5x ^{3}+3x−1)(x−1)**

**(iii) x−3(5+3x)**

**(iv) x ^{5 }(3−6x−9)**

**(iv) x−4(3−4x−5) **

**(vi) 2/x+1 − x ^{2 }/3x−1**

**Solution :**

(i)

(ii)

(iii)

(iv)

(v)

(vi)

**Question****10. Find the derivative of cos x from first principle.**

**Solution :**

,

**Question****11. Find the derivative of the following functions:**

**(i) sin x cos x **

**(ii) sec x **

**(iii) 5sec x + 4cos x**

**(iv) cosec x **

**(v) 3cot + 5cosec x**

**(vi) 5sin – 6cos x + 7 **

**(vii) 2tan x – 7sec x**

**Solution :**

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

### Miscellaneous Exercise

**Question****1. Find the derivative of the following functions from first principle:**

** (i) −x**

**( ii) (−x) ^{−1}**

**( iii) sin(x+1) **

**(iii)cos(x−π/8)**

**Solution :**

(i)Given:f(x) = -x then

(ii)Given:

then

(iii)Given:*f*(*x*) = sin (*x* + 1)

then

(iv)Given:

then

**Question****2. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + a)**

**Solution :**

f(x) = (x + a)

Accordingly,

By first principle,

**Question****3. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (px+q)(r/x+s)**

**Solution :**

Given:f(x) =

**Question****4. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):(ax+b)(cx+d) ^{2}**

**Solution :**

Given:f(x) = (ax+b)(cx+d)^{2}

**Question****5. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****6. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****7. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****8. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****9. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****10. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****11. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****12. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax+b) ^{n}**

**Solution :**

**Question****13. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):(ax+b) ^{n }(cx+d)^{m} **

**Solution :**

since,

**Question****14. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):sin (x + a) **

**Solution :**

**Question****15. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): cosec x cot x**

**Solution :**

**Question****16. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****17. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****18. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question**19. **Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sin ^{n}x **

**Solution :**

**Question****20. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****21. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****22. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x ^{4}(5 sin x – 3 cos x) **

**Solution :**

**Question****23. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x ^{2}+1)cosx**

**Solution :**

**Question****24. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax ^{2 }+ sinx)(p + qcosx)**

**Solution :**

**Question****25. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x+cosx) (x−tanx)**

**Solution :**

**Question****26. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****27. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****28. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**

**Question****29. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + sec x) (x – tan x)**

**Solution :**

let f(x) = (x + sec x) (x – tan x)

By product rule,

**Q****uestion****30. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):**

**Solution :**