**Question 1.Find the derivatives of the following w.r.t. x by using the method of the first principle.**

By first principle, we get

**(b) sin(3x)Solution:**

Let f(x) = sin 3x

f(x + h) = sin3(x + h) = sin(3x + 3h)

By first principle, we get

**(c) e ^{2x+1}Solution:**

**(d) 3 ^{x}Solution:**

**(e) log(2x + 5)Solution:**

Let f(x) = log(2x + 5)

∴ f(x + h) = log[2(x + h) + 5] = log (2x + 2h + 5)

By first principle, we get

**(f) tan(2x + 3)Solution:**

Let f(x) = tan(2x + 3)

∴ f(x + h) = tan[2(x + h) + 3] = tan(2x + 2h + 3)

By first principle, we get

**(g) sec(5x – 2)Solution:**

Let f(x) = sec(5x – 2)

f(x + h) = sec[5(x + h) – 2] = sec(5x + 5h – 2)

By first principle, we get

**(h) x√xSolution:**

**Question 2.Find the derivatives of the following w.r.t. x. at the points indicated against them by using the method of the first principle.Solution:**

Solution:

**(iii) 2 ^{3x-1} at x = 2Solution:**

**(iv) log(2x + 1) at x = 2Solution:**

Let f(x) = log(2x + 1)

∴ f(2) = log [2(2) + 1] = log 5 and

f(2 + h) = log [2(2 + h) + 1] = log(2h + 5)

By first principle, we get

**(v) e ^{3x-4} at x = 2Solution:**

Solution:

**Question 3.Show that the function f is not differentiable at x = -3,where f(x) = x**

^{2}+ 2 for x < -3= 2 – 3x for x ≥ -3Solution:

∴ L f'(-3) ≠ R f'(-3)

∴ f is not differentiable at x = -3.

**Question 4.Show that f(x) = x ^{2} is continuous and differentiable at x = 0.Solution:**

**Question 5.Discuss the continuity and differentiability of(i) f(x) = x |x| at x = 0Solution:**

Solution:

**Question 6.Discuss the continuity and differentiability of f(x) at x = 2.f(x) = [x] if x ∈ [0, 4). [where [ ] is a greatest integer (floor) function]Solution:**

Explanation:

x ∈ [0, 4)

∴ 0 ≤ x < 4

We will plot graph for 0 ≤ x < 4

not for x < 0 and upto x = 4 making on X-axis.

f(x) = [x]

∴ Greatest integer function is discontinuous at all integer values of x and hence not differentiable at all integers.

∴ f is not continuous at x = 2.

∵ f(x) = 1, x < 2

= 2, x ≥ 2

x ∈ neighbourhood of x = 2.

∴ L.H.L. = 1, R.H.L. = 2

∴ f is not continuous at x = 2.

∴ f is not differentiable at x = 2.

**Question 7.Test the continuity and differentiability off(x) = 3x + 2 if x > 2= 12 – x**

^{2}if x ≤ 2 at x = 2.Solution:

**Question 8.Solution:**

**Question 9.Examine the function= 0, for x = 0for continuity and differentiability at x = 0.Solution:**