Master Applications of Derivatives – Class 12 Maths Chapter 6 | Tangents, Max‑Min, Approximation, NCERT & PYQs

 Here's the complete guide to Chapter 6: Applications of Derivatives (Class 12 CBSE Maths, 2025–26 syllabus):

---

🧠 A. Key Concepts & Applications

1. Rate of Change

   • The derivative $dy/dx$ represents the rate at which $y$ changes w.r.t. $x$.

   • Applicable in physics (velocity, acceleration), economics (marginal cost/revenue), biology, etc.

2. Tangents and Normals

   • Slope of tangent at $x = a$: $m = f'(a)$.

   • Equation of tangent: $y - f(a) = f'(a)(x - a)$.

   • Normal line slope: $-1/f'(a)$.

3. Increasing/Decreasing Functions & Stationary Points

   • If $f'(x) > 0$ → increasing; $f'(x) < 0$ → decreasing.

   • Stationary points where $f'(x) = 0$.

   • Test using sign of $f'(x)$ or second derivative $f''(x)$ (if $>0$: min, if $<0$: max).

4. Maxima and Minima

   • Locally highest/lowest values.

   • Applications include geometry (maximizing area/volume) and optimization problems (cost, revenue).

5. Approximations via Derivatives

   • Use linearization: $\Delta y ≈ f'(a) \cdot \Delta x$.

   • Practical uses: estimating small errors, measuring slope near a point, etc.

---

📘 B. NCERT Exercise-Wise Detailed Solutions

1. Ex 6.1: Rate-of-change and tangent-line problems.

2. Ex 6.2: Find derivatives of complex functions using chain rule and apply in tangent equations.

3. Ex 6.3: Analyze increasing/decreasing intervals, locate stationary points.

4. Ex 6.4: Solve maxima/minima word problems (e.g., rectangular fencing, revenue).

5. Ex 6.5: Show differentiation-based approximations.

Each solution includes step-by-step:

1. Statement of the problem

2. Expression of $f'(x)$

3. Analysis and final result

---

🏆 C. CBSE Previous Year Questions (with Best Answers)

• 2023: Tangent to curve $y^2 = 4x + 3$ at point $(1,3)$.
  Solution: Differentiate implicitly → equation of tangent.

• 2022: Find maxima of function $f(x) = x^{1/3}(1 - x)^{2/3}$.
  Solution: Compute $f'(x)$, locate critical points, confirm using sign chart or $f''(x)$.

• 2021: A 20 m fencing problem to enclose a rectangular area with one side by a wall.
  Solution: Let width $x$, length $y$; express area $A(x)$; find $A'(x)=0$ → max.

#Tags:
#Class12Maths, #Calculus, #Derivatives, #MaximaMinima, #Tangents, #Optimization, #CBSEPYQs, #NCERTSolutions, #Chapter6Maths