Applications of Derivatives

Chapter 5: Continuity and Differentiability

Chapter 6 of the NCERT textbook, provides a definition of derivatives, rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, maxima and minima, first derivative test, maximum and minimum values of a function in a closed interval, and miscellaneous examples. Here, students can find the exercises explaining these concepts properly. Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima, and minima (the first derivative test is motivated geometrically, and the second derivative test is given as a provable tool).

Derivative for Rate of Change of a Quantity

Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. By using the application of derivatives we can find the approximate change in one quantity with respect to the change in the other quantity. Assume we have a function y = f(x), which is defined in the interval [a, a+h], then the average rate of change in the function in the given interval is

(f(a + h)-f(a))/h

Now using the definition of the derivative, we can write

f'(a)=limh→0f(a+h)−f(a)hf′(a)=limh→0f(a+h)−f(a)h

which is also the instantaneous rate of change of the function f(x) at a.

Now, for a very small value of h, we can write

f'(a) ≈ {(f(a+h) − f(a)}/h

f(a+h) ≈ {f(a) + f'(a)}/h

Approximation Value

Derivative of a function can be used to find the linear approximation of a function at a given value. The linear approximation method was given by Newton and he suggested finding the value of the function at the given point and then finding the equation of the tangent line to find the approximately close value to the function. The equation of the function of the tangent is

L(x) = f(a) + f'(a)(x−a)

Learn more about, Approximation

Tangent and Normal To a Curve

A tangent is a line to a curve that will only touch the curve at a single point and its slope is equal to the derivative of the curve at that point. The slope(m) of the tangent to a curve of a function y = f(x) at a point (x₁,y₁)(x₁,y₁) is obtained by taking the derivative of the function (m = f'(x) ).

By finding the slope of the tangent line to the curve and using the equation

m = (y₂−y₁)/(x₂−x₁)

we can find the equation of the tangent line to the curve. Similarly, we can find the equation of the normal line to the curve of a function at a point. This normal line will be normal(perpendicular) to the tangent line. Hence the slope of the normal line to a curve of a function y = f(x) at a point (x₁,y₁)(x₂,y₂) is given as follows.

n = -1/m = – 1/ f'(x)

By using the equation

−1/m = (y₂−y₁)/(x₂−x₁)

We can find the equation of the normal line to the curve.

Learn more about, Tangents and Normals

Maxima, Minima, and Point of Inflection

Maxima and minima are the peaks and valleys of a curve, whereas the point of inflection is the part of the curve where the curve changes its nature(from convex to concave or vice versa). We can find the maxima, minima, and point of inflection by using the first-order derivative test. According to this test, we first find the derivative of the function at a given point and equate it to 0, i.e., f'(c) = 0, (here we have found the slope of the curve equal to 0, which means it is a line parallel to the x-axis). Now if the function is defined in the given interval, then we check the value of f'(x) at the points lying to the left of the curve and to the right of the curve and check the nature of the f'(x), then we can say, that the given point is maxima or minima based on the below conditions.

Maxima when the slope or f’(x) changes its sign from +ve to -ve as we move via point c. And f(c) is the maximum value.
Minima when the slope or f’(x) changes its sign from -ve to +ve as we move via point c. And f(c) is the minimum value.
Point C is called the Point of inflection when the sign of slope or sign of the f’(x) doesn’t change as we move via c.

Increasing and Decreasing Functions

The increasing function is a function that seems to reach the top of the x-y plane whereas the decreasing function seems to reach the downside corner of the x-y plane. Let us say we have a function f(x) differentiable within the limits (a, b). Then we check any two points on the curve of the function.

If at any two points x1x1 and x2x2 such that x1x1 < x2x2, there exists a relation f(x1)f(x1) ≤ f(x2)f(x2), then the given function is increasing function in the given interval, and if f(x1)f(x1) < f(x2)f(x2), then the given function is strictly increasing function in the given interval.
And, if at any two points x1x1 and x2x2 such that x1x1 < x2x2, there exists a relation f(x1)f(x1) ≥ f(x2)f(x2), then the given function is decreasing function in the given interval and if f(x1)f(x1) > f(x2)f(x2), then the given function is strictly decreasing function in the given interval

Learn more about, Increasing and Decreasing Function

Class 12 Maths Formulas

Class 12 maths formulas page is designed for the convenience of the learners so that one can understand all the important concepts of Class 12 Mathematics directly and easily. Math formulae for Class 12 are for the students who find mathematics to be a nightmare and difficult to grasp. They may become hesitant and lose interest in studies as a result of this. We have included all of the key formulae for the 12th standard Maths topic, which students may simply recall, to assist them in understanding Maths in a straightforward manner. For all courses such as Integration, Differentiation, Trignometry, Relation and Functions, and so on, the formulae are provided here according to the NCERT curriculum.

Table of Content

Chapter 1: Relations and Functions
Chapter 2: Inverse Trigonometric Functions
Chapter 3: Matrices
Chapter 4: Determinants
Chapter 5: Continuity and Differentiability
Chapter 6: Applications of Derivatives
Chapter 7: Integrals
Chapter 8: Applications of Integrals
Chapter 9: Differential Equations
Chapter 10: Vector Algebra
Chapter 11: Three-Dimensional Geometry
Chapter 12: Linear Programming
Chapter 13: Probability