What is Application of Derivatives Class 12: Concepts & Examples

Definition and Importance of Application of Derivatives

The Application of Derivatives in Class 12 mathematics refers to using the derivative concept to solve practical problems involving change and optimisation. Derivatives represent how a function changes as its input changes, which is fundamental in physics, economics, and engineering.

In the NCERT syllabus, this chapter helps students:

• Understand rates of change in quantities
• Find slopes of tangents and normals to curves
• Determine maxima and minima values for functions
• Solve real-world problems involving optimisation

This chapter is vital for Class 12 students as it builds analytical thinking and problem-solving skills necessary for higher studies and competitive exams.

Understanding Rate of Change and Derivative Formulae

The derivative of a function $f(x)$ at a point $x$ is defined as the rate at which $f(x)$ changes with respect to $x$. Mathematically:

$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$

This formula helps calculate instantaneous rates such as velocity, growth rate, or slope of a curve.

Example: Find the derivative of $f(x) = x^2$.

$$ f'(x) = 2x $$

This means at any point $x$, the rate of change of $x^2$ is $2x$. For instance, at $x=3$, the slope is $6$.

Knowing these basics allows students to apply derivatives to various problems in the chapter.

Finding Tangents and Normals Using Derivatives

One key application of derivatives is finding the equations of tangents and normals to curves at given points.

• Tangent: A line that touches the curve at one point without crossing it.
• Normal: A line perpendicular to the tangent at the point of contact.

If the curve is $y = f(x)$ and the point of tangency is $(x_0, y_0)$, then:

• Slope of tangent, $m = f'(x_0)$
• Equation of tangent:

$$ y - y_0 = m (x - x_0) $$

• Equation of normal:

$$ y - y_0 = -\frac{1}{m} (x - x_0) $$

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

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

Maxima and Minima: Optimisation Using Derivatives

Derivatives help find local maxima and minima of functions, which is important for optimisation problems.

Steps to find maxima or minima:

1. Find $f'(x)$ and solve $f'(x) = 0$ to get critical points. 2. Use the second derivative test:

• If $f''(x) > 0$, then $f(x)$ has a local minimum at $x$.
• If $f''(x) < 0$, then $f(x)$ has a local maximum at $x$.

Example: Find maxima or minima of $f(x) = x^3 - 3x^2 + 4$.

• $f'(x) = 3x^2 - 6x$
• Set $f'(x) = 0$: $3x^2 - 6x = 0 \Rightarrow x( x - 2) = 0$ so $x=0$ or $x=2$
• $f''(x) = 6x - 6$
• At $x=0$: $f''(0) = -6 < 0$ (local max)
• At $x=2$: $f''(2) = 6 > 0$ (local min)

Thus, $f(x)$ has a local maximum at $x=0$ and a local minimum at $x=2$.

Applications in Real-Life Problems and Curve Sketching

The Application of Derivatives chapter also includes solving real-world problems such as:

• Finding the speed or acceleration from position-time functions
• Optimising area, volume, or cost in business and engineering
• Sketching curves by analysing increasing/decreasing intervals and points of inflection

Curve sketching involves:

This helps students visualise the behaviour of functions and apply derivatives effectively.

Summary of Key Formulas and Concepts

Here is a quick reference table for important formulas in Application of Derivatives:

Master these to solve most problems in Class 12 NCERT Application of Derivatives chapter.