Applications of Derivatives 📉

Derivative sirf calculate karna nahi — use karna bhi seekho! Rate of change, maxima, minima!

💰 Profit Maximize Karo!

Ek factory owner hai jisko apna profit maximize karna hai. Uska profit function P(x) = -x² + 100x - 200 hai (x = units produced). Ab yeh function kab maximum value lega? Derivative lao: P'(x) = -2x + 100. P'(x) = 0 par x = 50. Aur P''(x) = -2 < 0, matlab yeh maximum point hai! So 50 units produce karna sabse zyada profitable hai. Yeh sab applications of derivatives hai!

1. Rate of Change

dy/dx = rate of change of y with respect to x. Agar x time hai (t), toh dy/dt = rate of change of y over time.

✅ Example: Balloon Phool Raha Hai

Sphere ki radius 3 cm/sec ki rate se badh rahi hai. Volume ka rate of change find karo jab r = 5 cm.

V = (4/3)πr³ → dV/dt = 4πr² × dr/dt = 4π(25)(3) = 300π cm³/sec

2. Increasing & Decreasing Functions

Increasing: f'(x) > 0 on interval (a,b)
Decreasing: f'(x) < 0 on interval (a,b)
Critical points: f'(x) = 0 ya f'(x) undefined

✅ Find intervals where f(x) = x³ - 3x + 2 increasing/decreasing

f'(x) = 3x² - 3 = 3(x²-1) = 3(x-1)(x+1)

Critical points: x = -1, x = 1

x < -1: f'(x) > 0 → Increasing

-1 < x < 1: f'(x) < 0 → Decreasing

x > 1: f'(x) > 0 → Increasing

3. Tangents aur Normals

Slope of tangent at (x₀,y₀) = f'(x₀) = m Equation of tangent: y - y₀ = m(x - x₀) Slope of normal = -1/m (perpendicular to tangent) Equation of normal: y - y₀ = (-1/m)(x - x₀)

4. Maxima aur Minima

Local Maxima aur Minima Local Max f'=0, f''<0 Local Min f'=0, f''>0 Local Max
/* First Derivative Test */ At x=c where f'(c)=0: - If f' changes + to - : Local Maximum - If f' changes - to + : Local Minimum - If f' doesn't change sign: Neither (saddle/inflection) /* Second Derivative Test */ At x=c where f'(c)=0: - f''(c) < 0 → Local Maximum - f''(c) > 0 → Local Minimum - f''(c) = 0 → Inconclusive (use first test)

✅ Find maxima/minima of f(x) = x³ - 6x² + 9x + 15

f'(x) = 3x² - 12x + 9 = 3(x²-4x+3) = 3(x-1)(x-3)
Critical points: x=1 and x=3
f''(x) = 6x - 12. f''(1) = -6 < 0 → Local Max at x=1. f''(3) = 6 > 0 → Local Min at x=3

Local Max = f(1) = 1-6+9+15 = 19. Local Min = f(3) = 27-54+27+15 = 15

5. Practice Questions

Q1. A stone is dropped into pond. Radius of ripple increases at 5 cm/s. Rate of increase of area when r=10 cm?

Solution

A=πr². dA/dt=2πr×dr/dt=2π(10)(5)=100π cm²/s

Q2. Find the tangent line to y=x³-3x at (2,2).

Solution

y'=3x²-3. At x=2: slope=12-3=9. Tangent: y-2=9(x-2) → y=9x-16

Q3. Among all rectangles of perimeter 100m, find dimensions for maximum area.

Solution

Let l+w=50 (half perimeter). A=lw=l(50-l). dA/dl=50-2l=0 → l=25. Square with side 25m → max area=625m²