Application of Derivatives

Class 12 · Mathematics

Application of Derivatives

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Application of Derivatives

Derivative sirf slope nahi deta — uska istemaal real problems me hota hai: rates of change, increasing/decreasing behaviour, tangents, aur maximum–minimum values nikaalne me. Ye chapter wahi applications cover karta hai.

1. Rate of Change

\(\dfrac{dy}{dx}\) batata hai ki \(y\), \(x\) ke saath kitni tezi se badal raha hai. Agar dono \(x, y\) time \(t\) par depend karein, to \(\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}\) (chain rule).
Example

Circle ka area \(A = \pi r^{2}\). Radius \(2\) cm/s se badh raha hai. \(r = 5\) par area kitni tezi se badh raha? \[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 2\pi(5)(2) = 20\pi \ \text{cm}^2/\text{s}. \]

2. Increasing aur Decreasing Functions

Kisi interval par:
  • \(f'(x) > 0\) ⇒ function increasing.
  • \(f'(x) < 0\) ⇒ function decreasing.
  • \(f'(x) = 0\) ⇒ stationary (critical) point.
Example

\(f(x) = x^{2} - 4x\). \(f'(x) = 2x - 4\). \(f'(x) < 0\) jab \(x < 2\) (decreasing), \(f'(x) > 0\) jab \(x > 2\) (increasing).

3. Tangent aur Normal

Point \((x_1, y_1)\) par curve ki tangent ki slope \(= \left.\dfrac{dy}{dx}\right|_{(x_1, y_1)} = m\).

  • Tangent: \(y - y_1 = m(x - x_1)\).
  • Normal (tangent ke perpendicular): slope \(= -\dfrac{1}{m}\), so \(y - y_1 = -\dfrac{1}{m}(x - x_1)\).

4. Maxima aur Minima

First derivative test: critical point \(x = c\) par (\(f'(c) = 0\)):

  • \(f'\) ka sign \(+ \to -\) badle ⇒ local maximum.
  • \(f'\) ka sign \(- \to +\) badle ⇒ local minimum.

Second derivative test: \(f'(c) = 0\) ho to —

  • \(f''(c) < 0\) ⇒ local maximum.
  • \(f''(c) > 0\) ⇒ local minimum.
Example

\(f(x) = x^{2} - 4x + 3\) ka minimum. \(f'(x) = 2x - 4 = 0 \Rightarrow x = 2\). \(f''(x) = 2 > 0\) ⇒ minimum. \[ f(2) = 4 - 8 + 3 = -1. \] Toh minimum value \(-1\) hai \(x = 2\) par.

5. Steps for Word Problems (Optimization)

  1. Jise maximize/minimize karna hai use ek variable me likho.
  2. \(\dfrac{dy}{dx} = 0\) karke critical points nikaalo.
  3. Second derivative test se max ya min decide karo.
  4. Value calculate karo (aur domain/endpoints check karo).

Key Takeaways

  • \(\frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}\) — related rates.
  • \(f' > 0\) increasing, \(f' < 0\) decreasing.
  • Tangent slope \(= \frac{dy}{dx}\); normal slope \(= -\frac{1}{m}\).
  • Max/min: critical points par first ya second derivative test lagao.
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