All About Second Order 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 $f\left(x\right)=x^3$ is a function. Its first derivative will be f′(x) = $3x^2$ . 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.

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.

 

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.

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.

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.