Application of Derivatives

Edited By Team Careers360 | Updated on May 07, 2022 12:51 PM IST

Introduction:
The term "derivative" refers to something that is based on something else. A derivative is a mathematical term that expresses the rate of change of a function with respect to an independent variable. Derivatives are used in a variety of fields including mathematics, science, and engineering. The use of derivatives in estimating the rate of change of variables with regard to other quantities is a highly significant application. Without realising it, you employ such concepts on a daily basis! Your mother, for example, intuitively understands how much sugar to add to the tea to make it exactly the proper amount of sweet. By modeling real-life processes as functions of the known factors that affect them, we may get a better understanding of them. The next step in their study is to look at how they react when the factors change. Is it true that the functions grow or decrease? That is what we will learn about growing and decreasing functions in this lesson.

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List of topics according to NCERT and JEE Main/NEET syllabus:

Introduction to the application of derivatives
Rate of change of quantities
Approximation
Increasing and decreasing function
Maxima and minima
Tangent and normal

Important concepts and Laws:

If a quantity 'y' changes in response to a change in another variable 'x,' and an equation of the form y = f(x) is always fulfilled, indicating that 'y' is a function of 'x,' then the rate of change of 'y' with regard to 'x' is given by

ΔyΔx=y₂-y₁x₂-x₁

This is often referred to as the "Average Rate of Change."

The Instantaneous Rate of Change of a Function at a Given Point is defined as the rate of change of a function at a specific point, i.e. a specific value of 'x'. We have the definition of the derivative of a function at a point from the definition of the derivative of a function at a point.

Lim x → x₀ = y-y(x₀)x-x₀

Let y = f(x) be a differentiable function in interval x = (a,b). It means the function’s derivative exists at all points of its domain. Let’s take two points x1 and x2 in such a way that x1 < x2

If an inequality f(x1) < f(x2) satisfy for any two points x1 and x2 in the interval x, then the function f(x) is said to be rising in this interval.

Similarly, if an inequality f(x1) > f(x2) holds for any two points x1 and x2 in the interval x, then the function f(x) is considered declining in this interval.

Because the inequalities are tight, the functions are known as strictly growing or decreasing functions: f(x1) < f(x2) for strictly rising and f(x1) > f(x2) for strictly declining.

A tangent at a point on a curve is a straight line that touches the curve at that point and has the same slope as the curve's gradient/derivative. You may derive how to obtain the equation of the tangent to the curve at any point from the definition. The equation of the tangent to this curve at x = x0, given a function y = f(x)

A straight line that meets the curve at a point on the curve and is perpendicular to the tangent at that point is called a normal. If n is its slope, and m is the slope of the tangent at that point or the value of the gradient/derivative at that point, we obtain mn = -1.

NCERT Notes Subject wise link:

Importance of application of derivatives class 12

In Chapter 6, students learn about composite, implicit, logarithmic, inverse, trigonometric, and exponential functions, as well as their derivatives. The next stage in this course is to learn about derivatives and their applications in diverse disciplines. This topic is extremely useful for performing real-world analysis and interpreting graphical functions. Chapter 6 Application of Derivatives of Class 12 NCERT Solutions Mathematicians is well-crafted advisors who can apply a full comprehension of derivatives and their qualities. Finding turning points on a graph to determine the points where a function's maximum or minimum values occur, determining the intervals where a function increases or decreases, approximations, and errors are just a few of the key concepts and formulas covered in Class 12 Math’s Chapter 6 Application of Derivatives. These ideas are extensively illustrated through the use of examples and pictures.

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Frequently Asked Question (FAQs)

1. What can you deduce from the second derivative?

The second derivative reveals a great deal about the graph's qualitative behavior. The graph is concave up if the second derivative is positive at a given position. The critical point will be a local minimum if the second derivative is positive at the critical point. As a result, at an inflection point, the second derivative is zero.

2. In mathematics, why do we utilise approximation?

When the correct model is difficult to utilise, approximation refers to utilising a simpler technique or model. To make the computations easier, an approximation model might be employed. If insufficient knowledge prohibits the use of accurate demonstrations, approximations can be employed.

3. Can you explain the distinction between a tangent and a normal?

A tangent is a straight line that extends from a point on a curve and has a gradient equal to the curve's gradient at that point. A normal, on the other hand, is a straight line that extends from a curve's point in such a way that it is perpendicular to the point's tangent.

4. What are derivatives used for?

Because derivatives represent slope, we may use them to calculate the maxima and minima of a variety of functions. They can also be used to express at what rate a function is changing.

5. What is differential calculus and how is it used?

Differential calculus is a mathematical subdivision of calculus concerned with the study of the proportions at which values are altered. The derivatives of a function are the major objects of study in differential calculus.

6. What are the limits' applications?

A limit is a value that a function approaches as the input in mathematics.