What is Application of Derivatives Class 12: Definition & Uses

Understanding the Definition of Application of Derivatives

The application of derivatives involves using the derivative of a function to solve practical problems. In Class 12 NCERT Mathematics, this means applying the concept of differentiation to find:

• Rate of change of quantities
• Increasing and decreasing functions
• Local maxima and minima
• Tangents and normals to curves

A derivative, denoted as $\frac{dy}{dx}$, represents how a function $y=f(x)$ changes with respect to $x$. This fundamental idea helps students connect abstract calculus with real-world scenarios.

How Derivatives Help Identify Increasing and Decreasing Functions

One key application of derivatives is to determine where a function is increasing or decreasing:

• If $f'(x) > 0$ for an interval, the function is increasing there.
• If $f'(x) < 0$ for an interval, the function is decreasing there.

This helps in sketching graphs and understanding the behaviour of functions. For example, consider $f(x) = x^3 - 3x^2 + 2$:

$$f'(x) = 3x^2 - 6x$$

• For $x > 2$, $f'(x) > 0$ so $f$ is increasing.
• For $0 < x < 2$, $f'(x) < 0$ so $f$ is decreasing.

This analysis is vital for solving problems related to function behaviour in exams.

Finding Maxima and Minima Using Derivatives

Maxima and minima are points where a function reaches its highest or lowest value locally. To find these:

1. Find the critical points by solving $f'(x) = 0$. 2. Use the second derivative test:

• If $f''(x) > 0$, the point is a local minimum.
• If $f''(x) < 0$, the point is a local maximum.

Example:

For $f(x) = x^2 - 4x + 3$:

• $f'(x) = 2x - 4$; set $2x - 4 = 0$ gives $x=2$.
• $f''(x) = 2 > 0$ so $x=2$ is a minimum.

This method is widely tested in Class 12 exams for optimization problems.

Using Derivatives to Find Tangents and Normals to Curves

Derivatives help find the slope of a curve at any point, which is essential for equations of tangents and normals.

• The slope of the tangent at $x=a$ is $m = f'(a)$.
• Equation of tangent: $y - f(a) = f'(a)(x - a)$.
• Equation of normal: $y - f(a) = -\frac{1}{f'(a)}(x - a)$.

Example:

For $y = x^2$ at $x=1$:

• $f'(x) = 2x$, so $f'(1) = 2$.
• Tangent: $y - 1 = 2(x - 1)$ or $y = 2x - 1$.
• Normal: $y - 1 = -\frac{1}{2}(x - 1)$ or $y = -\frac{1}{2}x + \frac{3}{2}$.

This application is important for geometry-related questions in Class 12.

Real-Life Applications of Derivatives in Class 12 Mathematics

Derivatives are not just theoretical; they solve real-world problems such as:

• Optimizing profit and cost: Finding maximum profit or minimum cost using maxima/minima.
• Physics: Calculating velocity and acceleration as derivatives of displacement.
• Economics: Marginal cost and revenue analysis.
• Engineering: Designing curves and slopes.

These applications demonstrate the practical importance of the chapter and help students appreciate the usefulness of calculus.

Comparison of Key Concepts in Application of Derivatives

Here’s a quick comparison table summarizing important concepts:

This table helps students quickly revise key points before exams.