1.Ans: Limit must exist and equal the function value: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$.
3.Ans: LHL = $5$, RHL = $5$.
4.Ans: Yes, constant functions are continuous everywhere.
5.Ans: $(f+g)(x)$ is also continuous at $x=c$.
7.Ans: True. (Though it's not differentiable there).
9.Ans: Discontinuous at all integer values of $x$ (i.e., $x \in \mathbb{Z}$).
10.Ans: Yes, by the composition of continuous functions property.
11.Ans: Differentiability implies continuity. (If differentiable, then it must be continuous).
12.Ans: False. (e.g., $f(x)=|x|$ is continuous but not differentiable at 0).
13.Ans: No, it has a sharp corner (kink) at $x = 0$.
16.Ans: $u(x)v'(x) + v(x)u'(x)$.
17.Ans: $\frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$.
18.Ans: $\frac{dy}{dx} = 3x^2 + 5$.
19.Ans: $\frac{dy}{dx} = x\cos x + \sin x$. (Using Product Rule)
20.Ans: $\frac{dy}{dx} = \frac{\cos x(1) - x(-\sin x)}{\cos^2 x} = \frac{\cos x + x\sin x}{\cos^2 x}$.
21.Ans: $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.
22.Ans: $\frac{dy}{dx} = 2\cos(2x)$.
23.Ans: $\frac{dy}{dx} = -2x\sin(x^2)$.
24.Ans: $\frac{dy}{dx} = n(ax+b)^{n-1} \cdot a$.
25.Ans: $\frac{dy}{dx} = \frac{1}{2\sqrt{x^2+1}} \cdot 2x = \frac{x}{\sqrt{x^2+1}}$.
26.Ans: An equation where $y$ is not explicitly isolated on one side, but is mixed with $x$ (e.g., $f(x, y) = 0$).
27.Ans: $1 + \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -1$.
28.Ans: $\frac{dy}{dx} - \cos y \frac{dy}{dx} = 1 \Rightarrow \frac{dy}{dx} = \frac{1}{1-\cos y}$.
29.Ans: $2y \frac{dy}{dx}$.
30.Ans: $x\frac{dy}{dx} + y(1) = 0 \Rightarrow \frac{dy}{dx} = -\frac{y}{x}$.
31.Ans: $\frac{1}{\sqrt{1-x^2}}$.
32.Ans: $-\frac{1}{\sqrt{1-x^2}}$.
33.Ans: $\frac{1}{1+x^2}$.
34.Ans: $\frac{1}{|x|\sqrt{x^2-1}}$.
35.Ans: $-\frac{1}{1+x^2}$.
36.Ans: $\frac{2}{\sqrt{1-4x^2}}$.
37.Ans: $\frac{a}{a^2+x^2}$.
38.Ans: $-1$. (Simplify $\cos^{-1}(\sin x) = \frac{\pi}{2} - x$).
39.Ans: $x = \sin\theta$ or $x = \cos\theta$.
40.Ans: $x = \tan\theta$ or $x = \cot\theta$.
45.Ans: $\frac{1}{\sin x} \cdot \cos x = \cot x$.
46.Ans: When evaluating functions of the form $(Variable)^{(Variable)}$ like $x^x$, or for complex products and quotients.
47.Ans: Take the natural logarithm ($\log$) on both sides.
48.Ans: $\log u + \log v + \log w$.
49.Ans: $x^x(1 + \log x)$.
50.Ans: $1$. (Since $e^{\log_e x} = x$, its derivative is $1$).
51.Ans: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$.
52.Ans: $\frac{dx}{dt} = 2a$, $\frac{dy}{dt} = 2at$.
53.Ans: $\frac{dy}{dx} = \frac{2at}{2a} = t$.
54.Ans: $\frac{dy}{dx} = \frac{r\cos\theta}{-r\sin\theta} = -\cot\theta$.
55.Ans: $\frac{d^2y}{dx^2}$, which means differentiating the first derivative $\frac{dy}{dx}$ again with respect to $x$.
56.Ans: $\frac{dy}{dx} = 3x^2 \Rightarrow \frac{d^2y}{dx^2} = 6x$.
57.Ans: $\frac{dy}{dx} = \cos x \Rightarrow \frac{d^2y}{dx^2} = -\sin x$.
60.Ans: $0$. (First derivative is $m$, second is $0$).
61.Ans: 1. Continuous on $[a, b]$. 2. Differentiable on $(a, b)$. 3. $f(a) = f(b)$.
63.Ans: 1. Continuous on $[a, b]$. 2. Differentiable on $(a, b)$.
64.Ans: $f'(c) = \frac{f(b) - f(a)}{b - a}$.
65.Ans: There is at least one point where the tangent is parallel to the x-axis.
