Learn about the Second Order Derivative:

Let us take a step back and define what a first derivative is in order to better comprehend the second-order derivative. The first derivative is referred to as 

dy/dx 

reflects the pace at which the value of y changes in relation to the value of x Consider the following example: if a car travels 60 metres in 10 seconds, the speed is equal to the first order derivative of the distance travelled with respect to time (60 metres/second). The result is that the speed in this instance is stated as 60/10 m/s. 

We are all aware that speed fluctuates and does not remain constant indefinitely. As a result, the second derivative can be used to determine the rate at which the speed changes. 

Second Order Derivative: 

The second-order derivative of a function is nothing more than the derivative of the first derivative of the function under consideration. As a result, by calculating the second derivative, which is the rate of change in speed with respect to time, it is possible to determine the variation in speed of the car (the second derivative of distance travelled with respect to the time). 

Notation for second derivative: 

We have already seen Lagrange’s notation for the second derivative, denoted by the letter f”. 

The notation used by Leibniz for the second derivative is d²y/dx². Using the Leibniz notation, the second derivative of x³ + 2x² is written as d²/dx² (x³+2x²).  

Representation of second order derivative: 

On a graph, the first derivative reflects the slope of the function at a given point, while the second derivative describes how the slope varies as the independent variable in the graph is changed over time. The second derivative of a function with a changing slope can be used to explain the curvature of a graph such as the one shown. 

When looking at this graph, the blue line denotes the slope, which is the first derivative of the given function, as shown. In addition, the second derivative is utilised to define the nature of the function under consideration.. To find the maximum, minimum, or point of inflection, we can utilise the second derivative test, which is performed on the second derivative. 

If we look at it mathematically, 

y = f(x) 

Then, 

dy/dx = f’(x) 

Once it is determined that f’(x) can be differentiated, the process of differentiating begins. 

dy/dx 

Again, in relation to x, we obtain the second order derivative, i.e. 

d/dx (dy/dx) 

= d²y/dx² 

= f”(X) 

Higher order derivatives, on the other hand, can be defined in the same way as lower order derivatives. 

d³y/dx³ 

represents a third order derivative of the original, 

d⁴y/dx⁴ 

denotes a fourth order derivative, and so on, till the end.

In most cases, the second derivative of a given function correlates to the curvature or concavity of the graph being considered. If the value of the second-order derivative is positive, the graph of a function is upwardly concave, and vice versa. Whenever the value of the second-order derivative is negative, the graph of a function is said to be downwardly open.

As previously stated, the second derivative of a function determines the values of the local maximum and minimum, or inflexion points, of a function. These can be distinguished by the presence of the following conditions: 

• If f”(x) is greater than zero, the function f(x) has a local maximum at the point x.

• A local minimum for the function f(x) exists at the point x when f”(x) > 0.

• If f”(x) = 0, then it is not possible to draw any conclusions about the point x, which could be an inflexion point, from the data. 

Second Order Derivative examples: 

Let us look at an example to obtain a better understanding of second-order derivatives. 

Example 1: Find the value of d²y/dx² if y = e(x³)–3x⁴. 

Solution: Given that, y = e(x³)–3x⁴ 

When we differentiate this equation with respect to x, we get the following result: 

dy/dx = e(x³) x 3x² –12x³ 

Then, in order to determine the 2nd order derivative of the provided function, we differentiate the first derivative once more with respect to x, and so on. 

d²y/dx² = e(x³) x 3x² x 3x² + e(x³) x 6x – 36x² 

d²y/dx² = xe(x³) x (9x³ + 6) – 36x² 

This is the solution that is required. 

Example 2: Find the value of d²y/dx² if y = 4sin–1(x²). 

Solution: Given that, 

y = 4sin–1(x²) 

When we differentiate this equation with respect to x, we get the following result: 

dy/dx = 4/√(1–x⁴) x 2x 

Now, in order to determine the next higher order derivative of the provided function, we must differentiate the first derivative w.r.t. x once again. 

d²y/dx² = 2x x d/dx[4/√(1–x⁴)] + 4/√(1–x⁴) x d(2x)/dx 

[using d(uv)/dx = u (dv/dx) + v (du/dx)] 

d²y/dx² = –8(x⁴–1)/(x⁴–1)√(1–x⁴) 

Note: If we have a function f(x), we can use the formula to calculate the second order derivative (also known as second derivative) of the function f(x) using a single limit. 

f”x = limh→0 f(x+h)–2f(x)+f(x–h)/h² 

Conclusion: 

We may find out the rate at which a function of a real variable changes by taking the derivative of the function with respect to its argument. The derivative is denoted by the symbol dy/dx. The ratio dy/dx indicates the rate of change in y with regard to a given value of x. The slope of the tangent line to the graph of the function can also be used to define the derivative of a function. The derivative of the first-order derivative of the given function is referred to as the second-order derivative. It provides information about the shape of the graph as well as its concavity.