Math Class 12 JEE NEET Study Material pdf

Second Order Derivative | Explanation with Examples, Class 12 Math Notes Study Material Download Free PDF

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For understanding the second-order derivative, let us step back a bit and understand what a first derivative is. The first derivative dy/dx represents the rate of the change in y with respect to x. Considering an example, if the distance covered by a car in 10 seconds is 60 meters, then the speed is the first order derivative of the distance travelled with respect to time. Hence, the speed in this case is given as 60/10 m/s.

We know that speed also varies and does not remain constant forever. Thus, to measure this rate of change in speed, one can use the second derivative.

Second Order Derivative Definition and Representation

The second-order derivative is nothing but the derivative of the first derivative of the given function. So, the variation in speed of the car can be found out by finding out the second derivative, i.e. the rate of change of speed with respect to time (the second derivative of distance travelled with respect to the time).

Graphically the first derivative represents the slope of the function at a point, and the second derivative describes how the slope changes over the independent variable in the graph. For a function having a variable slope, the second derivative explains the curvature of the given graph.

Second Order Derivative

In this graph, the blue line indicates the slope, i.e. the first derivative of the given function. And the second derivative is used to define the nature of the given function. For example, we use the second derivative test to determine the maximum, minimum or the point of inflexion.

Mathematically, if  y = f(x) 

Then dy/dx = f'(x)

Now if f'(x) is differentiable, then differentiating dy/dx again w.r.t. x  we get 2nd order derivative, i.e.

\(\begin{array}{l} \frac {d}{dx} \left( \frac {dy}{dx} \right) = \frac {d^2y}{dx^2} = f”(x)\end{array} \)

Similarly, higher order derivatives can also be defined in the same way like d3y/dx3  represents a third order derivative, d4y/dx4 represents a fourth order derivative and so on.

Usually, the second derivative of a given function corresponds to the curvature or concavity of the graph. If the second-order derivative value is positive, then the graph of a function is upwardly concave. If the second-order derivative value is negative, then the graph of a function is downwardly open.

As it is already stated that the second derivative of a function determines the local maximum or minimum, inflexion point values. These can be identified with the help of below conditions:

  • If f”(x) < 0, then the function f(x) has a local maximum at x.
  • If f”(x) > 0, then the function f(x) has a local minimum at x.
  • If f”(x) = 0, then it is not possible to conclude anything about the point x, a possible inflexion point.

Second Order Derivative Examples

Let us see an example to get acquainted with second-order derivatives.

Example 1: If \(\begin{array}{l}y = e^{(x^3)} – 3x^4, \end{array} \)

Find second order derivative

Solution 1:

Given that,

\(\begin{array}{l}y = e^{(x^3)} – 3x^4, \end{array} \)

then differentiating this equation w.r.t. x we get,

\(\begin{array}{l} \frac {dy}{dx} = e^{(x^3)} ×3x^2 – 12x^3 \end{array} \)

Now to find the 2nd order derivative of the given function, we differentiate the first derivative again w.r.t. x,

\(\begin{array}{l} \frac {d^2y}{dx^2} = e^{(x^3)} × 3x^2 × 3x^2 + e^{(x^3)}  × 6x – 36x^2 \end{array} \)

\(\begin{array}{l}  \frac{d^2y}{dx^2} = xe^{(x^3)} × (9x^3 + 6 ) – 36x^2 \end{array} \)

This is the required solution.

Example 2: 

\(\begin{array}{l}\text{Find }\frac {d^2y}{dx^2} \text{ if y = 4 } sin^{-1}(x^2) \end{array} \)

Solution 2: 

Given that,

\(\begin{array}{l}y = 4 sin^{-1}(x^2), \end{array} \)

 then differentiating this equation w.r.t. x we get,

\(\begin{array}{l} \frac {dy}{dx} = \frac {4}{\sqrt{1 – x^4}} × 2x \end{array} \)

Now for finding the next higher order derivative of the given function, we need to differentiate the first derivative again w.r.t. x ,

\(\begin{array}{l} \frac {d^2y}{dx^2} = 2x × \frac {d}{dx}\left( \frac {4}{\sqrt{1 – x^4}}\right) + \frac {4}{\sqrt{1 – x^4}} \frac{d(2x)}{dx} \end{array} \)

\(\begin{array}{l}\text{(using }  \frac {d(uv)}{dx} = u \frac{dv}{dx} + v \frac {du}{dx})\end{array} \)

\(\begin{array}{l}\Rightarrow \frac {d^2y}{dx^2} = \frac {-8(x^4 + 1)}{(x^4 – 1)\sqrt{1 – x^4}} \end{array} \)

Note: We can also find the second order derivative (or second derivative) of a function f(x) using a single limit using the formula:

\(\begin{array}{l}f”(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}\end{array} \)

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.

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CBSE Class 12 Maths Syllabus 2025-26 with Marks Distribution

The table below shows the marks weightage along with the number of periods required for teaching. The Maths theory paper is of 80 marks, and the internal assessment is of 20 marks which totally comes out to be 100 marks.

CBSE Class 12 Maths Syllabus And Marks Distribution 2023-24

Max Marks: 80

No.UnitsMarks
I.Relations and Functions08
II.Algebra10
III.Calculus35
IV.Vectors and Three – Dimensional Geometry14
V.Linear Programming05
VI.Probability08
Total Theory80
Internal Assessment20
Grand Total100

Unit-I: Relations and Functions

1. Relations and Functions

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.

2. Inverse Trigonometric Functions

Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.

Unit-II: Algebra

1. Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

2. Determinants

Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

1. Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions like sin-1 x, cos-1 x and tan-1 x, derivative of implicit functions. Concept of exponential and logarithmic functions.
Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

2. Applications of Derivatives

Applications of derivatives: rate of change of quantities, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

3. Integrals 

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals

Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)

5. Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

dy/dx + py = q, where p and q are functions of x or constants.

dx/dy + px = q, where p and q are functions of y or constants.

Unit-IV: Vectors and Three-Dimensional Geometry

1. Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

2. Three – dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.

Unit-V: Linear Programming

1. Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit-VI: Probability

1. Probability

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.

Students can go through the CBSE Class 12 Syllabus to get the detailed syllabus of all subjects. Get access to interactive lessons and videos related to Maths and Science with ANAND CLASSES’S App/ Tablet.

Frequently Asked Questions on CBSE Class 12 Maths Syllabus 2025-26

Q1

Is Calculus an important chapter in the CBSE Class 12 Maths Syllabus?

Yes, Calculus is an important chapter in the CBSE Class 12 Maths Syllabus. It is for 35 marks which means that if a student is thorough with this chapter will be able to pass the final exam.

Q2

How many units are discussed in the CBSE Class 12 Maths Syllabus?

In the CBSE Class 12 Maths Syllabus, about 6 units are discussed, which contains a total of 13 chapters.

Q3

How many marks are allotted for internals in the CBSE Class 12 Maths syllabus?

About 20 marks are allotted for internals in the CBSE Class 12 Maths Syllabus. Students can score it with ease through constant practice.

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