📘 Chapter: Applications of Derivatives (Class 12 – NCERT Maths)
🔹 1. Introduction
Applications of Derivatives deal with the practical use of differentiation. This chapter helps us analyze how quantities change and how functions behave. It is widely used in mathematics, physics, economics, and engineering.
🔹 2. Rate of Change of Quantities
Derivatives are used to find the rate of change of one quantity with respect to another.
If y is a function of x, then:
dy/dx represents the rate of change of y with respect to x.
📌 Examples:
✔ Rate of change of distance with respect to time → velocity
✔ Rate of change of velocity with respect to time → acceleration
🔹 3. Increasing and Decreasing Functions 📈📉
A function f(x) is:
✔ Increasing in an interval if f′(x) > 0
✔ Decreasing in an interval if f′(x) < 0
✔ Constant if f′(x) = 0
📌 This concept helps in sketching graphs and analyzing function behaviour.
🔹 4. Tangents and Normals
The derivative of a function at a point gives the slope of the tangent at that point.
✔ Slope of tangent = dy/dx
✔ Slope of normal = −1 / (dy/dx)
Equations:
✔ Tangent: y − y₁ = m(x − x₁)
✔ Normal: y − y₁ = −1/m (x − x₁)
🔹 5. Maxima and Minima (Optimization Problems) ⭐
Maxima and minima deal with finding the maximum or minimum value of a function.
Necessary Condition
If f(x) has a maximum or minimum at x = a, then:
f′(a) = 0
Sufficient Condition
✔ If f′′(a) > 0 → minimum value at x = a
✔ If f′′(a) < 0 → maximum value at x = a
📌 If f′′(a) = 0, the test fails.
🔹 6. First Derivative Test
The first derivative test is used to find the nature of stationary points.
✔ If f′(x) changes from +ve to −ve → maximum
✔ If f′(x) changes from −ve to +ve → minimum
✔ If no change → neither maximum nor minimum
🔹 7. Second Derivative Test
Used when f′(a) = 0.
✔ f′′(a) > 0 → minimum
✔ f′′(a) < 0 → maximum
🔹 8. Applications of Maxima and Minima 🧠
Maxima and minima are used to:
✔ Minimize cost or time
✔ Maximize profit or area
✔ Solve real-life optimization problems
📌 Many board exam questions are based on word problems from this topic.
🔹 9. Approximations and Errors
Derivatives help in finding approximate values of quantities.
If y = f(x), then small change:
Δy ≈ dy = f′(x) dx
📌 Used in error analysis and measurement problems.
🔹 10. Summary of Important Formulae 📌
✔ Rate of change = dy/dx
✔ Increasing function: f′(x) > 0
✔ Decreasing function: f′(x) < 0
✔ Maximum/Minimum condition: f′(x) = 0
⭐ Key Takeaways
✔ Derivatives describe real-life changes
✔ Helps analyze graphs and motion
✔ Maxima–minima questions are scoring
✔ Very important chapter for board exams and competitive tests