Continuity and Differentiability
Class 11 me limits aur basic derivatives dekhe. Ab samjhenge ki ek function kahaan "tuta hua nahi" (continuous) hai, kahaan "smooth" (differentiable) hai, aur complex functions ko differentiate karne ke powerful rules.
1. Continuity
Function \(f\) point \(x = a\) par continuous hai agar: \[ \lim_{x \to a} f(x) = f(a). \] Yaani LHL \(=\) RHL \(=\) function ki actual value, teeno barabar hon.
Aasaan tareeke se: agar graph ko bina pen uthaaye banaya ja sake, to wo continuous hai.
\(f(x) = x^{2}\) har point par continuous hai, kyunki \(\lim_{x \to a} x^{2} = a^{2} = f(a)\).
2. Differentiability
Function \(x = a\) par differentiable hai agar wahan derivative exist kare: \[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}. \]
Important: agar function differentiable hai, to continuous bhi hoga. Par ulta zaroori nahi — \(f(x) = |x|\) \(x = 0\) par continuous hai par differentiable nahi (sharp corner).
3. Chain Rule
Composite function \(y = f(g(x))\) ke liye: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x). \]
\(y = \sin(x^{2})\). Outer \(\sin\), inner \(x^{2}\): \[ \frac{dy}{dx} = \cos(x^{2}) \cdot 2x = 2x\cos(x^{2}). \]
4. Standard Derivatives
\[ \frac{d}{dx}(x^{n}) = n x^{n-1}, \qquad \frac{d}{dx}(e^{x}) = e^{x}, \qquad \frac{d}{dx}(\ln x) = \frac{1}{x} \]
\[ \frac{d}{dx}(\sin x) = \cos x, \qquad \frac{d}{dx}(\cos x) = -\sin x, \qquad \frac{d}{dx}(\tan x) = \sec^{2}x \]
\[ \frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1 - x^{2}}}, \qquad \frac{d}{dx}(\tan^{-1}x) = \frac{1}{1 + x^{2}} \]
5. Special Differentiation Techniques
- Implicit: jab \(y\) ko \(x\) ke terms me alag nahi kiya ja sakta (jaise \(x^{2} + y^{2} = 25\)), dono taraf differentiate karke \(\frac{dy}{dx}\) solve karo.
- Logarithmic: \(y = [f(x)]^{g(x)}\) jaise functions ke liye pehle \(\ln\) lo, phir differentiate.
- Parametric: agar \(x = f(t)\), \(y = g(t)\), to \(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}\).
\(x^{2} + y^{2} = 25\) ko differentiate karo: \[ 2x + 2y\frac{dy}{dx} = 0 \;\Rightarrow\; \frac{dy}{dx} = -\frac{x}{y}. \]
6. Second Order Derivative
Derivative ka derivative = second order derivative:
\[ \frac{d^{2}y}{dx^{2}} = \frac{d}{dx}\!\left(\frac{dy}{dx}\right). \]
Key Takeaways
- Continuous: \(\lim_{x\to a} f(x) = f(a)\); differentiable ⇒ continuous (ulta zaroori nahi).
- Chain rule: \(\frac{dy}{dx} = f'(g(x))\,g'(x)\) — composite functions ki chaabi.
- Standard derivatives (trig, exp, log, inverse trig) yaad rakho.
- Implicit, logarithmic, parametric differentiation special cases handle karte hain.