Second Order Derivative

The first-order derivative is the derivative of a function determined by one variable or the derivative of a dependent variable with respect to an independent variable. The second-order derivative is a derivative at a point chosen by two variables. For instance, if you take the slope of a tangent line at some point, like in a graph, then that is a first-order derivative. A Second-Order Derivative enables us to comprehend the graph form of a function.

Second-Order Derivative Explained

To understand what a second-order derivative is, we must first understand what a derivative is. A derivative gives you the slope of a function at any point. A second-order derivative is a function’s derivative of its derivative. The first-order derivative is used to create it. So we get a function’s derivative first, then draw the derivative of the first derivative. A first-order derivative is denoted by f'(x) or dy/dx, whereas a second-order derivative is denoted by f”(x) or d2ydx2.

Concavity and inflexion points may be determined using a second-order derivative.

First-Order and Second-Order Derivatives

The derivative of the first derivative of a given function is the second-order derivative. So, the second derivative, or the rate of change of speed with respect to time, may be used to determine the variation in speed of an automobile (the second derivative of distance travelled with respect to time).

The first derivative graphically displays the function’s slope at a given location, whereas the second derivative explains how the slope varies when an independent variable in the graph changes. The second derivative of a function with a changing slope explains the curvature of the graph.

A graph’s curvature or concavity is usually represented by the second derivative of a function. The graph of a function is upwardly concave if the second-order derivative value is positive. The graph of a function is downwardly open if the second-order derivative value is negative.

Second-Order Derivative Examples: 

1. Given: y = log x, Find d2ydx2?

Answer:

Now as function, y = log x

Then dy/dx = d/dx is the first derivative (log x)

dy/dx = (1 / x) 

We’ll distinguish it even further to discover the second derivative,

d2ydx2 = ddxdydx 

             =ddx1x

             =-1x2                                                                     

1. Given: y =ex sin5x, Find d²y/dx²?

Answer:

Given that y =ex sin5x

The first derivative will therefore be:

dy/dx = d/dx (ex sin5x)

          = (multiplication rule)

          = ex (cos5x . 5) + sin5x . ex

          = ex (5cos5x + sin5x)

We’ll distinguish it even further to discover the second derivative,

d²y/dx² = d/dx (dy/dx)

             = d/dx (ex (5cos5x + sin5x))

             = ex(5(-sin5x)5 + 5cos5x) + (5cos5x + sin5x)(ex)

             = ex(10cos5x – 24sin5x)

             = 2ex(5cos5x – 12sin5x)

Second-Order Derivatives of a Parametric Function

When we first learned the derivative, we got the first derivative with respect to t. When we were asked to find the derivative of that with respect to t, we used the definition of the derivative again. That’s great for one-step functions, but for complex functions and equations, it’s not easy. We rely on the chain rule from calculus and write down the second derivative as:

If x = x(t) and y = y(t), then the second-order parametric form is:

First Derivative: dy/dx = (dy/dt) / (dx/dt)

Second Derivative: d²y/dx² = d/dx (dy/dx)

                                            = d/dt (dy/dt) / (dx/dt)

Remember: It is incorrect to express the preceding formula as d2y/dx2= (d2y/dt2) / (d²x/dt²), since this is not how the formula works.

The first derivative graphically displays the function’s slope at a given location, whereas the second derivative explains how the slope varies when the independent variable in the graph changes.

Second-Order Derivatives Graphically Represented

Generally speaking, there are two ways to measure a function’s slope: first and second derivatives. The first derivative tells us the value of the slope at a specific point. The first derivative is the slope of a curve at a specified point. In other words, the slope of the tangent line to a curve at that point. 

The second derivative shows how quickly the first derivative changes. If you’ve ever taken a calculus course, you’ll be familiar with this concept. In order to calculate the derivative, it’s necessary to first determine the equation of the curve. This can be done by mathematical or graphical methods. When calculating derivatives, you will be asked to take two derivatives – one in respect to time and one in respect to the money, sales, etc., depending on the context of your business problem.

When a function has a second derivative, the graph’s curvature or concavity may be seen in the graph. The graph of a function appears vertically concave if the second-order derivative coefficient is positive. The graph of a function is downwardly open if the second-order derivative value is negative. 

Concavity of Function

Allow f(x) to be a differentiable function in a convenient interval. The graph of f(x) may therefore be classified as follows:

Concave Up: If the y-value rises at a quicker and faster pace as you move from left to right, that segment of the curve is concave up.

Concave Down: This is the reverse of concave up, where the y-value drops from left to right, is termed concave down.

Inflection Points: Inflection locations are points when the function’s concavity changes, for example, from “concave up” to “concave down”.

The local maximum or lowest inflection point values are determined by a function’s second derivative. These can be recognised using the following criteria:

• If f”(x) < 0, then the function f(x) has a local maximum at x.

• The function f(x) has a local minimum at x if f”(x) > 0.

• If f”(x) = 0, it is impossible to draw any conclusions about the point x.

To clarify why the second derivative produces these results, a real-world comparison might be utilised. Consider a vehicle that accelerates quickly but initially has a negative acceleration. The position of the vehicle as the velocity approaches zero will clearly be the maximum distance from the starting place; beyond this interval, the velocity will turn negative, and the vehicle will reverse.

Conclusion

The derivative of the first derivative of a given function is the second-order derivative. The first derivative graphically displays the function’s slope at a given location, whereas the second derivative explains how the slope varies when an independent variable in the graph changes. The second derivative of a function with a changing slope explains the curvature of the graph.