Vardaan Institute - Mathematics Solutions

Section 1: Limits & Continuity Basics

Ans: $\lim_{x \to a^-} f(x)$

Ans: $\lim_{x \to a^+} f(x)$

Ans: RHL (Right Hand Limit)

Ans: $f(a)$ (Value of function at $a$)

Ans: $7$

Ans: $0$

Ans: True

10.

Ans: At $x = 0$

Section 2: Basic Differentiation Formulas

11.

Ans: $0$

12.

Ans: $nx^{n-1}$

13.

Ans: $1$

14.

Ans: $2x$

15.

Ans: $5x^4$

16.

Ans: $\frac{1}{2\sqrt{x}}$

17.

Ans: $-\frac{1}{x^2}$

18.

Ans: $\cos x$

19.

Ans: $-\sin x$

20.

Ans: $\sec^2 x$

21.

Ans: $-\text{cosec}^2 x$

22.

Ans: $\sec x \tan x$

23.

Ans: $-\text{cosec } x \cot x$

24.

Ans: $12x^3$

25.

Ans: $2x + \cos x$

26.

Ans: $5$

Section 3: Product, Quotient & Basic Chain Rules

27.

Ans: $v \cdot \frac{du}{dx}$

28.

Ans: $v^2$

29.

Ans: $\frac{dt}{dx}$

30.

Ans: $2\cos(2x)$

31.

Ans: $-3\sin(3x)$

32.

Ans: $5e^{5x}$

33.

Ans: $2(2x+1) \cdot 2 = 4(2x+1)$

34.

Ans: $4\sec^2(4x)$

35.

Ans: $x e^x + e^x = e^x(x+1)$

36.

Ans: $x\cos x + \sin x$

37.

Ans: $u = x$, and $v = \sin x$

Section 4: Implicit & Inverse Trigonometric Derivatives

38.

Ans: $\frac{dy}{dx}$

39.

Ans: $2y \frac{dy}{dx}$

40.

Ans: $\cos y \frac{dy}{dx}$

41.

Ans: $1 + \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -1$

42.

Ans: $2 + 3\frac{dy}{dx} = \cos x \Rightarrow \frac{dy}{dx} = \frac{\cos x - 2}{3}$

43.

Ans: $\frac{1}{\sqrt{1-x^2}}$

44.

Ans: $-\frac{1}{\sqrt{1-x^2}}$

45.

Ans: $\frac{1}{1+x^2}$

46.

Ans: $-\frac{1}{1+x^2}$

47.

Ans: $\frac{1}{x\sqrt{x^2-1}}$

48.

Ans: $-\frac{1}{x\sqrt{x^2-1}}$

Section 5: Exponential & Logarithmic Differentiation

49.

Ans: $e^x$

50.

Ans: $\frac{1}{x}$

51.

Ans: $a^x \log_e a$

52.

Ans: $-e^{-x}$

53.

Ans: $2^x \log_e 2$

54.

Ans: $\frac{1}{2x} \cdot 2 = \frac{1}{x}$

55.

Ans: $\log a + \log b$

56.

Ans: $\log a - \log b$

57.

Ans: $n \log m$

58.

Ans: Take the natural logarithm ($\log$) on both sides.

Section 6: Parametric & Second Order Derivatives

59.

Ans: $\frac{dy}{dt}$

60.

Ans: $2$

61.

Ans: $6t$

62.

Ans: $\frac{6t}{2} = 3t$

63.

Ans: $\frac{d^2y}{dx^2}$

64.

Ans: $2x$

65.

Ans: $2$

66.

Ans: $\frac{d}{dx}(3x^2) = 6x$

67.

Ans: $\frac{d}{dx}(5) = 0$

68.

Ans: $e^x$

Section 7: Mean Value Theorems Basics

69.

Ans: Closed

70.

Ans: Open

71.

Ans: $f(a) = f(b)$

72.

Ans: $0$

73.

Ans: No

74.

Ans: $b - a$

75.

Ans: X-axis

Vardaan Learning Institute