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The second derivative

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)).

Introduction

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 Figure. We see that near point \(A\) the value of \(f'(x)\) is positive and relatively close to zero, so near that point the graph is rising slowly. By contrast, near point \(B\), the derivative is negative and relatively large in absolute value, so \(f\) is decreasing rapidly near \(B\).

Besides asking whether the value of 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.

Exploration

Increasing or decreasing

So far, we have used the words increasing and decreasing intuitively to describe a function's graph. Here we define these terms more formally.

Simply put, an increasing function is one that is rising as we move from left to right along the graph, and a decreasing function is one that falls 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 if \(f'(x) > 0\) 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 \((a,b)\).

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 \[\begin{aligned}\end{aligned}\].

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 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 \(y = f'(x)\). In other words, just as the first derivative measures the rate at which the original function changes, the second derivative measures the rate at which the first derivative changes. The second derivative will help us understand how the rate of change of the original function is itself changing.

Concavity

In addition to asking whether a function is increasing or decreasing, it is also natural to inquire how a function is increasing or decreasing. There are three basic behaviors that an increasing function can demonstrate on an interval, as pictured in Figure: the function can increase more and more rapidly, it can increase at the same rate, or it can increase in a way that is slowing down. Fundamentally, we are beginning to think about how a particular curve bends, with the natural comparison being made to lines, which don't bend at all. More than this, we want to understand how the bend in a function's graph is tied to behavior characterized by the first derivative of the function.

On the leftmost curve in Figure, imagine drawing a sequence of tangent lines to the curve. As we move from left to right, the slopes of those tangent lines will increase. Therefore, the rate of change of the pictured function is increasing, and this explains why we sometimes say this function is increasing at an increasing rate. For the rightmost graph in Figure, observe that as \(x\) increases, the function increases, but the slopes of the tangent lines decrease. This function is increasing at a decreasing rate.

Similar options hold for how a function can decrease. Here we must be extra careful with our language, because decreasing functions involve negative slopes. Negative numbers present an interesting tension 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: \(x \gt y\) provided that \(x\) lies to the right of \(y\). So of course, \(-100\) is less than \(-2\). Informally, it can 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. The situation gets a bit more complicated when we think about how the slope of the tangent line to a decreasing function changes as we move from left to right.

We state these most recent observations formally as the definitions of the terms concave up and concave down.

Condensed — the full section is in Boelkins, Active Calculus.

Summary

A differentiable function \(f\) is increasing on an interval whenever its first derivative is positive, and decreasing 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.
A differentiable function is concave up whenever its first derivative is increasing (or equivalently whenever its second derivative is positive), and concave down whenever its first derivative is decreasing (or equivalently whenever its second derivative is negative). When a differentiable function is concave up on an interval, its tangent line always lies below the curve; when a differentiable function is concave down on an interval, its tangent line always lies above the curve. 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 \(y = -e^x\).
The units on the second derivative are units of output per unit of input per unit of input. Is is best not to try to simplify these units, as the second derivative's units and value tell us how the value of the derivative function is changing in response to changes in the input. In other words, the second derivative tells us the rate of change of the rate of change of the original function.

Practice (8)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

In the Desmos window below, you will find a graph of a function that you can experiment with by clicking on the point \(P\) on the graph, and moving it along the graph of \(f\). You will also see a green point representing the value of the second derivative of graph of \(f\) corresponding to the point \(P\). As you move the point \(P\), note that the tangent line to the graph moves along with the point. Then, do the following:
1. Carefully observe the shape of the graph and where it opens up versus down.
2. Investigate the relationship of the tangent line to the graph as you move the point. For example, at what points is the tangent line above or below the graph?
3. Notice the sign of the second derivative at various points, by observing the green dot corresponding to \(P\).
4. In the Desmos command line, toggle on the graph of the derivative function and repeat the steps above, paying attention to where the derivative is increasing and decreasing.
It will be useful to make a list of your observations as you experiment.

Drag each of the statements on the left onto the corresponding equivalent statement on the right. More than one statement on the left can correspond to a single statement on the right.
Finally, write a summary of your findings that will help you to understand how the sign of the second derivative relates to the shape of the graph.
Suppose that \(y = f(x)\) is a twice-differentiable function such that \(f''\) is continuous for which the following information is known: \(f(2) = -3\), \(f'(2) = 1.5\), \(f''(2) = -0.25\).
1. Is \(f\) increasing or decreasing near \(x = 2\)? Is \(f\) concave up or concave down near \(x = 2\)?
2. Do you expect \(f(2.1)\) to be greater than \(-3\), equal to \(-3\), or less than \(-3\)? Why?
3. Do you expect \(f'(2.1)\) to be greater than \(1.5\), equal to \(1.5\), or less than \(1.5\)? Why?
4. Sketch a graph of \(y = f(x)\) near \((2,f(2))\) and include a graph of the tangent line.
Разкрийте отговора
1. Since \(f'(2)\) is positive, \(f\) is increasing near \(x=2\), and since \(f''(2)\) is negative, \(f\) is concave down near \(x=2\).
2. We are given that \(f(2) = -3\). Since \(f\) is increasing at \(x=2\), we expect \(f(2.1)\) to be greater than \(f(2) = -3\).
3. Since \(f''(2)\) is negative, we also know that \(f'\) is decreasing near \(2\). Thus, we expect that \(f'(2.1)\) will be less than \(f'(2) = 1.5\).
4. In the following figure, we see a possible graph of a function \(f\) that passes through the point \((2,-3)\) at an instantaneous rate of \(f'(2) = 1.5\) and in such a way that its second derivative at \(x = 2\) is negative.
For a certain function \(y = g(x)\), its derivative is given by the function pictured in Figure.
1. What is the approximate slope of the tangent line to \(y = g(x)\) at the point \((2,g(2))\)?
2. How many real number solutions can there be to the equation \(g(x) = 0\)? Justify your conclusion fully and carefully by explaining what you know about how the graph of \(g\) must behave based on the given graph of \(g'\).
3. On the interval \(-3 \lt x \lt 3\), how many times does the concavity of \(g\) change? Why?
4. Use the provided graph to estimate the value of \(g''(2)\).
Разкрийте отговора
1. From the given graph of \(g'\), we may estimate that \(g'(2) \approx 1.4\).
2. Observe that \(g'\) is always positive. This tells us that \(g\) must be always increasing. Therefore, it follows that \(g\) may cross the \(x\)-axis at most one time, and hence there can be at most one solution to \(g(x) = 0\).
3. On \(-3 \lt x \lt 3\), \(g\) changes concavity \(g\) times, since \(g'\) changes from increasing to decreasing or from decreasing to increasing \(9\) times. Whenever \(g'\) is increasing, \(g''\) is positive, and whenever \(g'\) is decreasing, \(g''\) is negative. Hence, whenever \(g'\) changes from increasing to decreasing or vice versa, it follows that \(g''\) changes sign, and this causes a change in the concavity of \(g\). We therefore see that while \(g\) is always increasing, \(g\) changes concavity many times.
4. From the given graph of \(g'\), we can see that \(g'(1.9) \approx 0.9\) and \(g'(2.1) \approx 2\). Using a central difference, it follows that \[\begin{aligned}\end{aligned}\].
For each prompt that follows, sketch a possible graph of a function on the interval \(-3 \lt x \lt 3\) that satisfies the stated properties.
1. \(y = f(x)\) such that \(f\) is increasing on \(-3 \lt x \lt 3\), concave up on \(-3 \lt x \lt 0\), and concave down on \(0 \lt x \lt 3\).
2. \(y = g(x)\) such that \(g\) is increasing on \(-3 \lt x \lt 3\), concave down on \(-3 \lt x \lt 0\), and concave up on \(0 \lt x \lt 3\).
3. \(y = h(x)\) such that \(h\) is decreasing on \(-3 \lt x \lt 3\), concave up on \(-3 \lt x \lt -1\), neither concave up nor concave down on \(-1 \lt x \lt 1\), and concave down on \(1 \lt x \lt 3\).
4. \(y = p(x)\) such that \(p\) is decreasing and concave down on \(-3 \lt x \lt 0\) and is increasing and concave down on \(0 \lt x \lt 3\).
Разкрийте отговора
1. When a function is increasing, its graph rises as \(x\) increases, and when a function is concave up, that means the graph of the function looks bowl-shaped. A graph of an increasing, concave up function \(f\) on the interval \((-3,0)\) is shown in the following figure. When a function is concave down, its graph looks like an upside down bowl. The function \(f\) in the figure is increasing and concave down on the interval \((0,3)\) Note that \(f\) is also increasing on the entire interval \((-3,3)\).
2. When a function is increasing, its graph rises as \(x\)increases, and when a function is concave down, that means the graph of the function looks like an upside down bowl. A graph of an increasing, concave down function \(g\) on the interval \((-3,0)\) is shown in the figure below. When a function is concave up, its graph looks like a bowl. The function \(g\) in the figure is increasing and concave up on the interval \((0,3)\). Note that \(g\) is also increasing on the entire interval \((-3,3)\).
3. When a function is decreasing, its graph falls as \(x\) increases, and when a function is concave up, that means the graph of the function looks bowl-shaped. A graph of a decreasing, concave up function \(h\) on the interval \((-3,-1)\)is shown in the figure below. To be neither concave up or concave down, a graph has to be linear, so the graph of \(h\) in the following figure is a straight line on \((-1,1)\). When a function is concave down, its graph looks like an upside down bowl. The function \(h\) in the figure is decreasing and concave down on the interval \((1,3)\). Note that \(h\) is decreasing on the entire interval \((-3,3)\).
4. When a function is decreasing, its graph falls as \(x\) increases, and when a function is concave down, that means the graph of the function looks like an upside down bowl. A graph of a decreasing, concave down function \(p\) on the interval \((-3,0)\) is shown in the figure below. When a function is increasing, its graph rises as \(x\) increases. The function \(p\) in the figure is increasing and concave down on the interval \((0,3)\).
Suppose that \(y = f(x)\) is a twice-differentiable function such that \(f''\) is continuous for which the following information is known: \(f(2) = -3\), \(f'(2) = 1.5\), \(f''(2) = -0.25\).
1. Is \(f\) increasing or decreasing near \(x = 2\)? Is \(f\) concave up or concave down near \(x = 2\)?
2. Do you expect \(f(2.1)\) to be greater than \(-3\), equal to \(-3\), or less than \(-3\)? Why?
3. Do you expect \(f'(2.1)\) to be greater than \(1.5\), equal to \(1.5\), or less than \(1.5\)? Why?
4. Sketch a graph of \(y = f(x)\) near \((2,f(2))\) and include a graph of the tangent line.
Разкрийте отговора
1. Since \(f'(2)\) is positive, \(f\) is increasing near \(x=2\), and since \(f''(2)\) is negative, \(f\) is concave down near \(x=2\).
2. We are given that \(f(2) = -3\). Since \(f\) is increasing at \(x=2\), we expect \(f(2.1)\) to be greater than \(f(2) = -3\).
3. Since \(f''(2)\) is negative, we also know that \(f'\) is decreasing near \(2\). Thus, we expect that \(f'(2.1)\) will be less than \(f'(2) = 1.5\).
4. In the following figure, we see a possible graph of a function \(f\) that passes through the point \((2,-3)\) at an instantaneous rate of \(f'(2) = 1.5\) and in such a way that its second derivative at \(x = 2\) is negative.
For a certain function \(y = g(x)\), its derivative is given by the function pictured in Figure.
1. What is the approximate slope of the tangent line to \(y = g(x)\) at the point \((2,g(2))\)?
2. How many real number solutions can there be to the equation \(g(x) = 0\)? Justify your conclusion fully and carefully by explaining what you know about how the graph of \(g\) must behave based on the given graph of \(g'\).
3. On the interval \(-3 \lt x \lt 3\), how many times does the concavity of \(g\) change? Why?
4. Use the provided graph to estimate the value of \(g''(2)\).
Разкрийте отговора
1. From the given graph of \(g'\), we may estimate that \(g'(2) \approx 1.4\).
2. Observe that \(g'\) is always positive. This tells us that \(g\) must be always increasing. Therefore, it follows that \(g\) may cross the \(x\)-axis at most one time, and hence there can be at most one solution to \(g(x) = 0\).
3. On \(-3 \lt x \lt 3\), \(g\) changes concavity \(g\) times, since \(g'\) changes from increasing to decreasing or from decreasing to increasing \(9\) times. Whenever \(g'\) is increasing, \(g''\) is positive, and whenever \(g'\) is decreasing, \(g''\) is negative. Hence, whenever \(g'\) changes from increasing to decreasing or vice versa, it follows that \(g''\) changes sign, and this causes a change in the concavity of \(g\). We therefore see that while \(g\) is always increasing, \(g\) changes concavity many times.
4. From the given graph of \(g'\), we can see that \(g'(1.9) \approx 0.9\) and \(g'(2.1) \approx 2\). Using a central difference, it follows that \[\begin{aligned}\end{aligned}\].
For each prompt that follows, sketch a possible graph of a function on the interval \(-3 \lt x \lt 3\) that satisfies the stated properties.
1. \(y = f(x)\) such that \(f\) is increasing on \(-3 \lt x \lt 3\), concave up on \(-3 \lt x \lt 0\), and concave down on \(0 \lt x \lt 3\).
2. \(y = g(x)\) such that \(g\) is increasing on \(-3 \lt x \lt 3\), concave down on \(-3 \lt x \lt 0\), and concave up on \(0 \lt x \lt 3\).
3. \(y = h(x)\) such that \(h\) is decreasing on \(-3 \lt x \lt 3\), concave up on \(-3 \lt x \lt -1\), neither concave up nor concave down on \(-1 \lt x \lt 1\), and concave down on \(1 \lt x \lt 3\).
4. \(y = p(x)\) such that \(p\) is decreasing and concave down on \(-3 \lt x \lt 0\) and is increasing and concave down on \(0 \lt x \lt 3\).
Разкрийте отговора
1. When a function is increasing, its graph rises as \(x\) increases, and when a function is concave up, that means the graph of the function looks bowl-shaped. A graph of an increasing, concave up function \(f\) on the interval \((-3,0)\) is shown in the following figure. When a function is concave down, its graph looks like an upside down bowl. The function \(f\) in the figure is increasing and concave down on the interval \((0,3)\) Note that \(f\) is also increasing on the entire interval \((-3,3)\).
2. When a function is increasing, its graph rises as \(x\)increases, and when a function is concave down, that means the graph of the function looks like an upside down bowl. A graph of an increasing, concave down function \(g\) on the interval \((-3,0)\) is shown in the figure below. When a function is concave up, its graph looks like a bowl. The function \(g\) in the figure is increasing and concave up on the interval \((0,3)\). Note that \(g\) is also increasing on the entire interval \((-3,3)\).
3. When a function is decreasing, its graph falls as \(x\) increases, and when a function is concave up, that means the graph of the function looks bowl-shaped. A graph of a decreasing, concave up function \(h\) on the interval \((-3,-1)\)is shown in the figure below. To be neither concave up or concave down, a graph has to be linear, so the graph of \(h\) in the following figure is a straight line on \((-1,1)\). When a function is concave down, its graph looks like an upside down bowl. The function \(h\) in the figure is decreasing and concave down on the interval \((1,3)\). Note that \(h\) is decreasing on the entire interval \((-3,3)\).
4. When a function is decreasing, its graph falls as \(x\) increases, and when a function is concave down, that means the graph of the function looks like an upside down bowl. A graph of a decreasing, concave down function \(p\) on the interval \((-3,0)\) is shown in the figure below. When a function is increasing, its graph rises as \(x\) increases. The function \(p\) in the figure is increasing and concave down on the interval \((0,3)\).

Symbols used here

Euler's number
2.71828…, the base whose exponential is its own derivative.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\sin,\ \cos,\ \tan

sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

\lim_{x \to a} f(x)

limit
The value f(x) approaches as x approaches a.

f'(x),\ \frac{dy}{dx}

derivative
Instantaneous rate of change; slope of the graph.

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

y',\ y''

first and second derivative of y
Prime notation for derivatives with respect to x (or t).

C,\ C_1,\ C_2

arbitrary constants
Constants of integration fixed by initial conditions.

How to: The second derivative

How does the derivative of a function tell us whether the function is increasing or decreasing on an interval?
What can we learn by taking the derivative of the derivative (the second derivative) of a function f?
What does it mean to say that a function is concave up or concave down? How are these characteristics connected to certain properties of the derivative of the function?
What are the units of the second derivative? How do they help us understand the rate of change of the rate of change?

Questions people ask

What is a derivative in one sentence?

The slope of the graph at a point — the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate — the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function — a polynomial times an exponential, log or trig function.

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Parts of this page are adapted from Boelkins, Active Calculus (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.

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