The Second Derivative Calculator is a free online tool that displays the second-order derivative for a given function. STUDYQUERIES’ online second derivative calculator tool makes the calculation faster and displays the second-order derivative within a fraction of a second.
How to Use the Second Derivative Calculator?
The procedure to use the second derivative calculator is as follows:
- Step 1: Enter the function in the appropriate input field
- Step 2: To obtain the derivative, click “Submit”
- Step 3: In the output field, you will find the second-order derivative of a function
Second Derivative Calculator
What Is Second Derivative And Calculator?
In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of f. Generally, the second derivative measures how a quantity’s rate of change is itself changing; for example, the second derivative of an object’s position with respect to its position in time is its instantaneous acceleration or its rate of velocity change over time. Leibniz’s notation:
$$a=\mathbf{\frac{dv}{dt}}=\mathbf{\frac{d^2x}{dt^2}}$$
where \(a\) is acceleration, \(v\) is velocity, \(t\) is time, \(x\) is position, and \(d\) is the instantaneous “delta” or change. The last expression \(\mathbf{\frac{d^2x}{dt^2}}\) is the second derivative of position \((x)\) with respect to time.
Let’s understand in a more simple way
The second derivative corresponds to the curve or concavity of the graph of a function. The graph of a function with a positive second derivative is concave upward, while the graph of a function with negative second derivative curves in the opposite direction.
Given a differentiable function \(y=f(x)\), we know that its derivative, \(y=f′(x)\), is a related function whose output at \(x=a\) tells us the slope of the tangent line to \(y=f(x)\) at the point \((a,f(a))\). That is, heights on the derivative graph tell us the values of slopes on the original function’s graph.
At a point where \(f′(x)\) is positive, the slope of the tangent line to f is positive. Therefore, on an interval where \(f′(x)\) is positive, the function \(f\) is increasing (or rising). Similarly, if \(f′(x)\) is negative on an interval, the graph of \(f\) is decreasing (or falling).
The derivative of \(f\) tells us not only whether the function \(f\) is increasing or decreasing on an interval, but also how the function \(f\) is increasing or decreasing. Look at the two tangent lines shown in the Figure above.
We see that near point A the value of \(f′(x)\) is positive and relatively close to zero, and near that point, the graph is rising slowly. By contrast, near point \(B\), the derivative is negative and relatively large in absolute value, and \(f\) is decreasing rapidly near \(B\).
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In addition to asking whether the derivative function is positive or negative and whether it is large or small, we can also ask “how is the derivative changing?”.
Because the derivative, \(y=f′(x)\), is itself a function, we can consider taking its derivative — the derivative of the derivative — and ask “what does the derivative of the derivative tell us about how the original function behaves?” We start with an investigation of a moving object.
Let’s took an example
\(\mathbf{\color{red}{The\ position\ of\ a\ car\ driving\ along\ a\ straight\ road\ at\ time\ t\ in\ minutes\ is\ given\\ by\ the\ function\ y=s(t)\ that\ is\ pictured\ in\ Figure.\ The\ car’s\ position\ function\ has\\ units\ measured\ in\ thousands\ of\ feet.\ For\ instance,\ the\ point\ (2,4)\ on\ the\ graph\\ indicates\ that\ after\ 2\ minutes,\ the\ car\ has\ traveled\ 4000\ feet.}}\)
- In everyday language, describe the behavior of the car over the provided time interval. In particular, you should carefully discuss what is happening on each of the time intervals \([0,1],\ [1,2],\ [2,3],\ [3,4],\ and\ [4,5]\), plus provide commentary overall on what the car is doing on the interval \([0,12]\).
- On the lefthand axes provided in Figure, sketch a careful, accurate graph of \(y=s′(t)\).
- What is the meaning of the function \(y=s′(t)\) in the context of the given problem? What can we say about the car’s behavior when \(s′(t)\) is positive? when \(s′(t)\) is zero? when \(s′(t)\) is negative?
- Rename the function you graphed in (b) to be called \(y=v(t)\). Describe the behavior of \(v\) in words, using phrases like “v is increasing on the interval ……” and “v is constant on the interval ….….”
- Sketch a graph of the function \(y=v′(t)\) on the righthand axes provide in Figure. Write at least one sentence to explain how the behavior of \(v′(t)\) is connected to the graph of \(y=v(t)\).
Increasing Or Decreasing
Until now, we have used increasing and decreasing intuitively to describe a function’s graph. This section defines them formally.
Given a function \(f(x)\) defined on the interval \((a,b)\), we say that ff is increasing on \((a,b)\) provided that for all \(x, y\) in the interval \((a,b)\), if \(x \lt y\), then \(f(x) \lt f(y)\). Similarly, we say that \(f\) is decreasing on \((a,b)\) provided that for all \(x, y\) in the interval \((a,b)\), if \(x \lt y\), then \(f(x) \gt f(y)\).
According to the graph, an increasing function is one that is rising as we move from left to right, and a decreasing function is one that is falling as the value of the input increases. If the function has a derivative, the sign of the derivative tells us whether the function is increasing or decreasing.
Let \(f\) be a function that is differentiable on an interval \((a,b)\). It is possible to show that that if \(f′(x) \gt 0\) for every \(x\) such that \(a \lt x \lt b\), then \(f\) is increasing on \((a,b)\); similarly, if \(f′(x) \lt 0\) on \((a,b)\), then \(f\) is decreasing on \((a,b)\).
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For example, the function pictured in Figure is increasing on the entire interval \(−2 \lt x \lt 0\), and decreasing on the interval \(0 \lt x \lt 2\). Note that the value \(x=0\) is not included in either interval since, at this location, the function is changing from increasing to decreasing.
The Second Derivative
We are now accustomed to investigating the behavior of a function by examining its derivative. The derivative of a function \(f\) is a new function given by the rule
$$\mathbf{f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}}$$
Because \(f′\) is itself a function, it is perfectly feasible for us to consider the derivative of the derivative, which is the new function \(y=[f′(x)]′\). We call this resulting function the second derivative of \(y=f(x)\) and denote the second derivative by \(y=f′′(x)\). Consequently, we will sometimes call \(f′\) “the first derivative” of \(f\), rather than simply “the derivative” of \(f\).
The second derivative is defined by the limit definition of the derivative of the first derivative. That is,
$$\mathbf{f”(x)=\lim_{h \to 0}\frac{f'(x+h)-f'(x)}{h}}$$
The meaning of the derivative function still holds, so when we compute \(y=f′′(x)\), this new function measures slopes of tangent lines to the curve \(y=f′(x)\), as well as the instantaneous rate of change of \(y=f′(x)\).
The second derivative measures the rate at which the first derivative changes, just as the first derivative measures the rate at which the original function changes. We can understand how the rate of change of the original function is changing by taking the second derivative.
Concavity
In addition to asking whether a function is increasing or decreasing, it is also natural to ask how a function is changing. An increasing function can demonstrate three basic behaviors, as shown in Figure: it can increase more quickly, it can increase at the same rate, or it can increase in a slow manner.
In essence, we are starting to think about how a particular curve bends, with the comparison being made to lines, which do not bend at all. But we also wish to understand how the bend in a function’s graph relates to the behavior characterized by the first derivative of the function.
All three functions are increasing, but at increasing rates, constant rates, and decreasing rates, respectively.
Draw tangent lines to the curve on the leftmost curve in the Figure above. From left to right, the slopes of those tangent lines will increase. In other words, the rate of change of the pictured function is increasing, which explains why we say that the function is increasing at an increasing rate.
For the rightmost graph in the Figure above, observe that as \(x\) increases, the function increases, but the slopes of the tangent lines decrease. This function is increasing at a decreasing rate.
The same applies to how a function can decrease. The language we use here must be extra careful since decreasing functions involve negative slopes. Negative numbers present a fascinating conflict between common language and mathematical language.
For example, it can be tempting to say that “−100 is bigger than −2.” But we must remember that “greater than” describes how numbers lie on a number line: x>y provided that x lies to the right of y. So of course, −100 is less than −2. Informally, it might be helpful to say that “−100 is more negative than −2.” When a function’s values are negative, and those values get more negative as the input increases, the function must be decreasing.
Three functions are all decreasing in the figure below, but they are decreasing in different ways.
In Figure, the middle graph clearly shows a function decreasing at a constant rate. On the first curve, draw a series of tangent lines. As we move from left to right, the slopes of these lines become less negative. Thus, even though the values of the first derivative are negative, they are increasing, so we can say that the curve on the left is decreasing.
Thus, we are left with only the rightmost curve in the Figure. For that function, the slopes of the tangent lines are negative throughout the pictured interval, but as we move from left to right, the slopes become even more negative. Thus, the slope of the curve is decreasing, and we say that the function is decreasing at a decreasing rate.
We now introduce the notion of concavity which provides simpler language to describe these behaviors. When a curve opens upward on a given interval, like the parabola \(y=x^2\) or the exponential growth function \(y=e^x\), we say that the curve is concave up on that interval. Likewise, when a curve opens down, like the parabola \(y=−x^2\) or the opposite of the exponential function \(y=−e^x\(, we say that the function is concave down. Concavity is linked to both the first and second derivatives of the function.
As shown in the figure below, we see two functions and tangents to each of them. On the left-hand plot, where the function is concave up, note how the tangent lines always lie below the curve itself, and how the slopes of the tangent lines increase from left to right.
In other words, the function f is concave up on the interval shown because its derivative, \(f′\), is increasing on that interval. On the righthand plot in Figure, where the function is concave down, we see that the tangent lines are always above the curve, and the slope of the tangent lines is decreasing as we move from left to right. The fact that its derivative, \(f′\), is decreasing makes \(f\) concave down on the interval.
At left, a function that is concave up; at right, one that is concave down. We state these most recent observations formally as the definitions of the terms concave up and concave down.
Let \(f\) be a differentiable function on an interval \((a,b)\). Then f is concave up on \((a,b)\) if and only if \(f′\) is increasing on \((a,b)\); \(f\) is concave down on \((a,b)\) if and only if \(f′\) is decreasing on \((a,b)\).
Conclusion
- A differentiable function f increases on an interval whenever its first derivative is positive and decreases whenever its first derivative is negative.
- By taking the derivative of the derivative of a function \(f\), we arrive at the second derivative, \(f”\). The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to \(f\) is increasing or decreasing.
- Differentiable functions are concave up when their first derivative is increasing (or equivalently when their second derivative is positive) and concave down when their first derivative is decreasing (or equivalently when their second derivative is negative). Examples of functions that are everywhere concave up are \(y=x^2\) and \(y=e^x\); examples of functions that are everywhere concave down are \(y=−x^2\) and \(y=−e^x\).
- Units on the second derivative are “units of output per unit of input per unit of input.” They tell us how the derivative function changes as the input changes. In other words, the second derivative tells us how much the original function changed over time.
FAQs
Why is it written \(\frac{d^2y}{dx^2}\)?
The first derivative is denoted as \(\frac{dy}{dx}\). The second derivative is a derivative, with respect to \(x\), of the first. Therefore we denote it as \(\frac{d}{dx}\frac{dy}{dx}\). When the brackets are opened, we remain with \(\frac{d^2y}{dx^2}\).
What is the meaning of the second derivative?
The second derivative is the rate of change of a point at a graph (the “slope of the slope” if you will). This can be used to find the acceleration of an object (velocity is given by the first derivative).
What is the derivative of \(e^{2x}\)?
2e^{2x}
What is the first and second derivative?
While the first derivative can tell us if the function is increasing or decreasing, the second derivative. tells us if the first derivative is increasing or decreasing.
Why do we use the second derivative?
The derivative tells us if the original function is increasing or decreasing. Because f′ is a function, we can take its derivative. The second derivative gives us a mathematical way to tell how the graph of a function is curved. The second derivative tells us if the original function is concave up or down.