All About Second Order Derivative

Continuity and Differentiability 2025 ( Maths Continuity and Differentiability )

Jaya Sharma
Updated on Aug 25, 2025 11:49 IST

By Jaya Sharma, Assistant Manager - Content

The second-order derivative measures how rate of change of a function is itself changing. It tells you how slope of function is changing as well as the concavity of the function and acceleration. In simple words, it is the derivative of the first derivative.

second order derivative

This is an important topic of class 12 continuity and differentiability chapter that explains what second derivatives are and what they explain. Once you have completed this and other topics, you can start NCERT excercise of Continuity and Differentiability.

Table of content
  • What is the Second Derivative?
  • Applications of Second Order Derivative
  • Properties of Second Order Derivative
  • Graphical Representation of Second-Order Derivatives
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What is the Second Derivative?

Mathematically, if f(x) is the function, then its first derivative will be f′(x) and the second derivative will be f′′(x). It tells you whether the graph of a function is curving upward or downward. If  f′′(x)>0, the function is concaving up and if f′′(x)<0, then the function is concaving down.

Let us consider a function and find its second order derivative. Suppose f ( x ) = x 3 is a function. Its first derivative will be f′(x) = 3 x 2 . The second derivative in this case will be f′′(x) = 6x. At x = 1, f′′(1)=6, which means the function will concave up at x = 1.

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Applications of Second Order Derivative

CBSE board covers questions based on the application of second-order derivative to understand how much does the student know what this term means. The second derivative has different applications in different fields, as mentioned below:

  • Physics: The second-order derivative describes acceleration, which is the rate of change of velocity over time.
  • Engineering: The second-order derivative analyses the stability and structural integrity of systems, including determining bending moment in beams.
  • Economics: The second-order derivative is used for studying the rate of change of marginal costs or revenue. This derivative provides insights into how quickly the cost or profits will increase or decrease. 
  • Mathematics: Second-order derivatives reveal the concavity of a function, which indicates whether a graph will curve upward or downward. 
  • Biology: The second derivative can model the rate of change of growth rates, like how the population of a species accelerates or decelerates over time. This helps in understanding ecological dynamics and in predicting future population trends.
  • Finance: Second derivative can help in measuring the convexity of bonds that indicates how duration of bond changes in response to fluctuations in the interest rate.

 

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Properties of Second Order Derivative

The NEET exam or JEE Main exam will cover questions around the properties of the second order derivative. Second-order derivative has several important mathematical properties that analyze the behaviour of functions, curves and other real-world phenomena. Let us take a look at each one of them:

1. The second order derivative determines the concavity of the graph of a function. Concavity helps in visualizing how a curve bends and how it sketches graphs accurately.

  • If f''(x) > 0, the function will concave up on that interval.
  • In case, f"(x)<0, the function will concave down on that interval.

2. The second derivative test classifies critical points (where f'(x) = 0) as local maxima, local minima or at neither point. 

  • If f′(c) = 0 and f′′(c)>0, then f has local minima at x=c.
  • If f′(c) = 0 and f′′(c)<0, then f has local maxima at x=c.
  • If f"(c) = 0, then the test will be inconclusive which means further analysis will be required.

3. Using second order derivative of the position function, it is possible to determine acceleration in physics. 

4. Second derivative is related to curvature of curve which indicates how sharp a curve will bend at a given point.

5. For linear functions, the second derivative will always be zero since the slope is constant and does not change.

6. Second order derivative is the foundation to understand higher-order derivatives.

7. In order for the second derivative to exist at a point, the first derivative must be differentiable at that point.

Q&A Icon
Commonly asked questions
Q:  

Let S = { 1 , 2 , 3 , 4 } .  Then the number of elements in the set  is……………….

{ f : S × S : f i s o n t o a n d f ( a , b ) = f ( b , a ) a ( a , b ) S × S }

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A: 

  ( 1 , 1 ) ( 1 , 4 ) ( 4 , 1 ) ( 2 , 4 ) ( 4 , 2 ) ( 3 , 4 ) ( 4 , 3 ) ( 4 , 4 )  all have only one image.

(2, 1) (1, 2), (2, 2) each element has 3 choice.

(3, 2) (2, 3) (3, 1) (1, 3) (3, 3) each element has two choices.

total function = 3 × 3 × 2 × 2 × 2 = 72

Case I

None of the pre image have 3 as image, total functions = 2 × 2 × 1 × 1 × 1 = 4

Case II

None of the pre images have 2 as image then number of function = 25 = 32

Case III

None of the pre image have either 3 or 2 as image

Total function = 15 = 1

Total number of onto function

= 72 – 4 – 32 + 1 = 37

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Graphical Representation of Second-Order Derivatives

The first derivative represents slope of a function at a point, whereas the second order derivative describe the changes in slope over the independent variable in the graph.

second derivative

Questions based on second order derivative will be asked in IIT JAM exam and IISER entrance exam. In the graph, the line labelled as slope is the first derivative of given function. Second derivative of given function corresponds to curvature or concavity of graph. When second-order derivative is positive, the graph of the function is concave upwards. On the other hand, if the second order derivative value is negative, the graph will be curved downwards.

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