Continuity and Differentiability Class 12 Formulas

1. Continuity

Definition of Continuity

A function \(f(x)\) is continuous at \(x = a\) if:

\[\lim_{x \to a} f(x) = f(a)\]

Conditions for Continuity at \(x = a\)

\(f(a)\) exists (function is defined at \(x = a\))
\(\lim_{x \to a} f(x)\) exists
\(\lim_{x \to a} f(x) = f(a)\)

Left and Right Hand Limits

Left Hand Limit:

\[\lim_{x \to a^-} f(x) = \lim_{h \to 0} f(a - h)\]

Right Hand Limit:

\[\lim_{x \to a^+} f(x) = \lim_{h \to 0} f(a + h)\]

Important: \(\lim_{x \to a} f(x)\) exists if and only if \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)\)

2. Differentiability

Definition of Derivative

\[f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\]

Alternative form:

\[f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}\]

Left and Right Hand Derivatives

Left Hand Derivative:

\[f'_-(a) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h}\]

Right Hand Derivative:

\[f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}\]

Function is differentiable at \(x = a\) if: \(f'_-(a) = f'_+(a)\)

3. Relationship between Continuity and Differentiability

If \(f(x)\) is differentiable at \(x = a\), then \(f(x)\) is continuous at \(x = a\)
Converse is not true: Continuity does not imply differentiability
Example: \(f(x) = |x|\) is continuous at \(x = 0\) but not differentiable

4. Standard Derivatives

Basic Functions

\(\frac{d}{dx}(c) = 0\) where \(c\) is constant
\(\frac{d}{dx}(x^n) = nx^{n-1}\)
\(\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}\)
\(\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}\)

Exponential and Logarithmic Functions

\(\frac{d}{dx}(e^x) = e^x\)
\(\frac{d}{dx}(a^x) = a^x \ln a\)
\(\frac{d}{dx}(\ln x) = \frac{1}{x}\)
\(\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}\)

Trigonometric Functions

\(\frac{d}{dx}(\sin x) = \cos x\)
\(\frac{d}{dx}(\cos x) = -\sin x\)
\(\frac{d}{dx}(\tan x) = \sec^2 x\)
\(\frac{d}{dx}(\cot x) = -\csc^2 x\)
\(\frac{d}{dx}(\sec x) = \sec x \tan x\)
\(\frac{d}{dx}(\csc x) = -\csc x \cot x\)

Inverse Trigonometric Functions

\(\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}\)
\(\frac{d}{dx}(\cos^{-1} x) = \frac{-1}{\sqrt{1-x^2}}\)
\(\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}\)
\(\frac{d}{dx}(\cot^{-1} x) = \frac{-1}{1+x^2}\)
\(\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}\)
\(\frac{d}{dx}(\csc^{-1} x) = \frac{-1}{|x|\sqrt{x^2-1}}\)

5. Rules of Differentiation

Basic Rules

Sum/Difference Rule:

\[\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\]

Constant Multiple Rule:

\[\frac{d}{dx}[cf(x)] = c \cdot f'(x)\]

Product Rule

\[\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)\]

Quotient Rule

\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}\]

Chain Rule

If \(y = f(g(x))\), then:

\[\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\]

Or in Leibniz notation:

\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\] where \(u = g(x)\)

6. Parametric Differentiation

If \(x = f(t)\) and \(y = g(t)\), then:

\[\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}\]

7. Implicit Differentiation

For equations of the form \(F(x,y) = 0\):

Differentiate both sides with respect to \(x\)
Treat \(y\) as a function of \(x\)
Use chain rule: \(\frac{d}{dx}(y^n) = ny^{n-1} \frac{dy}{dx}\)
Solve for \(\frac{dy}{dx}\)

8. Logarithmic Differentiation

For functions of the form \(y = [f(x)]^{g(x)}\):

Take natural logarithm: \(\ln y = g(x) \ln f(x)\)
Differentiate both sides
\(\frac{1}{y} \frac{dy}{dx} = g'(x) \ln f(x) + g(x) \frac{f'(x)}{f(x)}\)
\(\frac{dy}{dx} = y \left[g'(x) \ln f(x) + g(x) \frac{f'(x)}{f(x)}\right]\)

9. Higher Order Derivatives

Notation

First derivative: \(f'(x)\) or \(\frac{dy}{dx}\)
Second derivative: \(f''(x)\) or \(\frac{d^2y}{dx^2}\)
nth derivative: \(f^{(n)}(x)\) or \(\frac{d^ny}{dx^n}\)

Leibniz Rule for nth Derivative of Product

\[(uv)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} u^{(k)} v^{(n-k)}\]

10. Mean Value Theorems

Rolle's Theorem

If \(f(x)\) is continuous on \([a,b]\), differentiable on \((a,b)\), and \(f(a) = f(b)\), then there exists \(c \in (a,b)\) such that:

\[f'(c) = 0\]

Mean Value Theorem (Lagrange)

If \(f(x)\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there exists \(c \in (a,b)\) such that:

\[f'(c) = \frac{f(b) - f(a)}{b - a}\]

11. L'Hôpital's Rule

For indeterminate forms \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\):

\[\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\]

(provided the limit on the right exists)

12. Important Limits

\(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
\(\lim_{x \to 0} \frac{\tan x}{x} = 1\)
\(\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}\)
\(\lim_{x \to 0} (1 + x)^{1/x} = e\)
\(\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e\)
\(\lim_{x \to 0} \frac{e^x - 1}{x} = 1\)
\(\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1\)
\(\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a\)