### Video Transcript

Estimating Derivatives

In this video, we will learn how to use numerical and graphical methods to estimate the derivative of a function. We’ll be looking at examples of how both of these methods work. We will start by looking at the numerical method for estimating a derivative.

In these types of questions, we’ll be asked to estimate 𝑓 prime of 𝑎 for some 𝑎, where 𝑓 prime is the derivative of 𝑓 with respect to 𝑥. We’ll also be given some 𝑓 of 𝑥 values for 𝑥-values around 𝑎 and including 𝑎. We will be using these 𝑓 of 𝑥 values along with their corresponding 𝑥-values in order to estimate the derivative at 𝑎. And the way in which we’ll do this is to use the fact that the derivative function is the slope function. And so, to estimate the derivative, we simply estimate the slope of the function. We’ll now cover the method of how we do this.

So if we’re trying to find the derivative of 𝑓 at 𝑎 and we’ve been given the values of 𝑓 of 𝑎, 𝑓 of 𝑏, and 𝑓 of 𝑐, where 𝑏 is less than 𝑎 which is less than 𝑐, then we can use these three 𝑓 values along with the three 𝑥-values to estimate the slope on either side of 𝑎. We know that the slope is equal to the change in 𝑦 over the change in 𝑥. We can use our 𝑓 values for the change in 𝑦 and the corresponding 𝑥-values for the change in 𝑥. Using these values, we can estimate the slope on the left and right of 𝑎. Now, remember, these are only estimates since we’re using the slope of a straight line in order to estimate the slope of a curve. We can say that the slope on the left of 𝑎 is roughly equal to 𝑓 of 𝑎 minus 𝑓 of 𝑏 over 𝑎 minus 𝑏. And the slope on the right of 𝑎 is roughly equal to 𝑓 of 𝑐 minus 𝑓 of 𝑎 over 𝑐 minus 𝑎.

Now that we have estimated the slope on both the left and right of 𝑎, we can estimate the slope at 𝑎 by taking the mean of these two slopes. We can find the mean by adding these two slopes together and dividing by two. Therefore, we form an equation for estimating the derivative of 𝑓 at 𝑎. It is that 𝑓 prime of 𝑎 is roughly equal to 𝑓 of 𝑎 minus 𝑓 of 𝑏 over 𝑎 minus 𝑏 plus 𝑓 of 𝑐 minus 𝑓 of 𝑎 over 𝑐 minus 𝑎 all over two. Now, this estimate for 𝑓 prime of 𝑎 will be more accurate the closer the values of 𝑏 and 𝑐 are to 𝑎. However, it’s crucial that our values of 𝑏 and 𝑐 are on either side of 𝑎.

Let’s now look at an example of how this method can be used.

Given that 𝑦 is equal to 𝑓 of 𝑥 is a function for four known values, where 𝑓 of two is equal to three, 𝑓 of six is equal to 3.75, 𝑓 of seven is equal to four, and 𝑓 of 11 is equal to 4.25, estimate 𝑓 prime of seven.

In this question, we’ve been asked to estimate the derivative of 𝑓 at seven and we’ve been given 𝑓 of 𝑥 values near seven. Therefore, we can use the numerical method in order to estimate this derivative. We have that 𝑓 prime of 𝑎 is roughly equal to 𝑓 of 𝑎 minus 𝑓 of 𝑏 over 𝑎 minus 𝑏 plus 𝑓 of 𝑐 minus 𝑓 of 𝑎 over 𝑐 minus 𝑎 all over two, where 𝑏 is less than 𝑎, which is less than 𝑐. And we want to choose the closest possible values to 𝑎 for 𝑏 and 𝑐. In our case, since we’re trying to find 𝑓 prime of seven, 𝑎 is equal to seven. And the closest 𝑥-values on either side of seven, for which we’ve been given their 𝑓 values, is six and 11. So we can let six be equal to 𝑏 and 11 be equal to 𝑐.

Next, we can simply substitute these values into our formula. We have that 𝑓 prime of seven is roughly equal to 𝑓 of seven minus 𝑓 of six over seven minus six plus 𝑓 of 11 minus 𝑓 of seven over 11 minus seven all over two. Now, we know the values of 𝑓 of six, 𝑓 of seven, and 𝑓 of 11 since they’ve been given to us in the question. So we can substitute these values in, which leaves us with this. And next, we can simplify the fractions in the numerator to give us 0.25 over one plus 0.25 over four all over two.

Now, we can write 0.25 over one as 0.25. And we can write 0.25 over four as one-fourth multiplied by 0.25. Next, we can rewrite the 0.25s as one-fourth. And next, we can multiply through and then add the two fractions in the numerator and then finally divide five over 16 by two to give us that 𝑓 prime of seven is roughly equal to five over 32. Our solution can also be written in decimal form as 𝑓 prime of seven is approximately equal to 0.15625.

Now, before we move on to the next example, let’s take a closer look at what we’re actually doing here. This graph shows the function 𝑓 of 𝑥. Let’s say a question has asked us to estimate the derivative of 𝑓 at 𝑎. Now, in this question, we’re not actually given this graph of 𝑓 of 𝑥. However, we have been given the value of 𝑓 of a. And we’ve also been given two other 𝑓 values: 𝑓 of 𝑏 and 𝑓 of 𝑐 such that 𝑏 is less than 𝑎 and 𝑐 is greater than 𝑎. When using our numerical method for estimating the derivative at 𝑎, we are estimating the slope to the left of 𝑎 by finding the slope of the straight line going from the point 𝑏, 𝑓 of 𝑏 to 𝑎, 𝑓 of 𝑎.

And we’re doing a similar method to estimate the slope to the right of 𝑎. We’re finding the slope between the points 𝑎, 𝑓 of 𝑎 and 𝑐, 𝑓 of 𝑐. And these are our calculations of 𝑓 of 𝑎 minus 𝑓 of 𝑏 over 𝑎 minus 𝑏 and 𝑓 of 𝑐 minus 𝑓 of 𝑎 over 𝑐 minus 𝑎. Then, in our final step of estimating the derivative, we’re simply taking the mean of these two slopes. And that’s what gives us our estimate for 𝑓 prime of 𝑎.

Let’s now move on to our second example involving a table.

Use the table to estimate 𝑓 prime of six.

Here, we’ve been asked to estimate the derivative of 𝑓 at six. And we’ve been given some 𝑥-values along with their corresponding 𝑓 values around six and also including six. Therefore, we can use our formula for the numerical method for estimating a derivative, which tells us that the derivative of 𝑓 at 𝑎 is roughly equal to the mean of the slopes on the left and right of 𝑎. Now, for this formula, 𝑏 must be less than 𝑎 which is less than 𝑐. And we must choose values of 𝑏 and 𝑐 as close to 𝑎 as possible such that 𝑏 and 𝑐 are on either side of 𝑎.

In our case, the value of 𝑎 is six and the closest 𝑥-value to the left and right of six, which we’ve been given, are four and eight. Therefore, four is equal to 𝑏 and eight is equal to 𝑐. Now that we have our values for 𝑎, 𝑏, and 𝑐, we can simply substitute them into our formula. We have that 𝑓 prime of six is roughly equal to 𝑓 of six minus 𝑓 of four over six minus four plus 𝑓 of eight minus 𝑓 of six over eight minus six all over two. Next, we can substitute in the values for 𝑓 of four, 𝑓 of six, and 𝑓 of eight since they’ve been given to us in the table.

After substituting in these values, we’re able to simplify the fractions in the numerator, since 4.25 minus 3.9 is equal to 0.35, 4.8 minus 4.25 is equal to 0.55, and six minus four and eight minus six are both equal to two. Next, we can divide both fractions in the numerator by the denominator. So that’s two, giving us 0.35 over four plus 0.55 over four. Since we have a common denominator here, we can add these two fractions together, which gives us 0.9 over four. Simplifying this fraction, we reach our solution, which is that the estimate for the derivative of 𝑓 at six is 0.225.

Now, let’s move on to see how we can estimate a derivative using a graph. If we have a function 𝑓 of 𝑥, as shown here, then we can estimate the derivative of 𝑓 at some point 𝑎 by again using the fact that the derivative is the slope function. And we can estimate the slope at any point on a graph by drawing an approximate tangent at that point and then finding the slope of that tangent. So in order to find the slope of 𝑓 at 𝑎, we simply draw an approximate tangent and then find the slope of that tangent. And this approximate slope of 𝑓 at 𝑎 is our estimate for the derivative at 𝑎.

Let’s see how we can do this in the following example.

For the given graph, estimate 𝑓 prime of three.

Here, we have been asked to estimate the derivative of 𝑓 at three. And we’ve been given a graph of 𝑓 of 𝑥. Now, our graphical method for estimating a derivative is to draw a tangent at the point where we’re trying to find the derivative and to find the slope of that tangent. Now, if you find the value of three on our 𝑥-axis and look at its point on our graph of 𝑓 of 𝑥, we can see that a tangent has already been drawn for this point. And so, what we need to do in order to estimate the derivative of 𝑓 at three is to find the slope of this tangent. The most accurate way to find the slope of this tangent is to pick the two points furthest apart on our tangent, which we can accurately read of our axes.

We can see that we have a point at five, seven. And we also have another point on our tangent at one, negative one. Therefore, we can use these two points in order to find the slope of this tangent. We use the fact that the slope of a line is equal to the change in 𝑦 over the change in 𝑥. A change in 𝑦 for our tangent is the difference in the 𝑦-values of the points which we’ve found. So that’s seven minus negative one. And the change in 𝑥 is the difference in the 𝑥-values for these same points. So that’s five minus one. Now, let’s remember that it’s important to put the corresponding points on the same side. So the five and the seven came from the point five, seven and they both go on the left. Whereas the one and the negative one came from the point one, negative one, and they both go on the right.

Now, it doesn’t matter which way around we put these points as long as they’re consistent with one another. For example, this fraction is also equal to negative one minus seven over one minus five since the negative one and the one are both on the left and the seven and the five are both on the right. We can simplify this fraction in order to find the slope of our tangent is equal to eight over four, which is, of course, equal to two.

Now that we found the slope of our tangent, we can use this to estimate 𝑓 prime of three since the derivative is the slope function. And we’ve found the slope of the tangent at three. This gives us a solution that our estimate for the derivative of 𝑓 at three is two. Now, the reason why this is an estimate and not an accurate answer is because we do not know how accurate the tangent is at the point three. Now, in this previous example, we saw how we can use a tangent to find an estimate for the slope and therefore the derivative at a point. However, the tangent on the graph is often not given to us. So we’ll need to draw in a tangent for ourselves, as we’ll see in our final example.

For the given graph, estimate 𝑓 prime of negative 1.5.

Here, we’ve been asked to estimate the derivative of 𝑓 at negative 1.5. And we’ve been given a graph of 𝑓 of 𝑥. Now, in order to estimate this derivative using the graph, we’ll be using the fact that the derivative is the slope function. Therefore, when we were asked to estimate the derivative of 𝑓 at negative 1.5, we’re also being asked to estimate the slope of 𝑓 at negative 1.5. And one way which we can estimate the slope of 𝑓 at any point is to draw a tangent at that point and find the slope of the tangent. Since we’ve been asked to estimate the derivative at 𝑥 is equal to negative 1.5 which we’ll draw a tangent on 𝑓 at 𝑥 is equal to negative 1.5. Negative 1.5 is here on our graph and we can see the corresponding point on 𝑓 of 𝑥.

Now, we simply draw a tangent to 𝑓 at this point. And this is what our tangent should look like. Now, our estimate tangent here is in fact a horizontal line. And because of this, we can say that the slope of our tangent must be equal to zero. This is because a slope is equal to the change in 𝑦 over the change in 𝑥. And we can see that as 𝑥 gets larger or smaller, the value of 𝑦 for this horizontal line will remain the same. Therefore, there’s no change in 𝑦. And so, the change in 𝑦 over the change in 𝑥 will be equal to zero. This is true for any horizontal line. Now that we’ve found the slope for our tangent, we can use this to estimate the derivative. We can say that the derivative of 𝑓 at negative 1.5 is approximately zero.

Let’s now look at one final example for this video.

For the given graph, estimate 𝑓 prime of 0.5.

We have been asked to estimate the derivative of 𝑓 at 0.5. And we’ve been given the graph of 𝑓. Since the derivative is the slope function, we simply need to estimate the slope of 𝑓 at 0.5. And we can do this by drawing a tangent for 𝑓 at 0.5. We can find the point on 𝑓 of 𝑥, which corresponds to the 𝑥-value of 0.5. Then, we can draw a tangent to the graph at this point. We want to try and draw the tangent as long as possible so we can get an accurate value for its slope.

We can look at the end points of the tangent we have drawn, which are here and here. And the coordinates of these points are 2.6, five and negative five, negative five. Along with the fact that the slope of a straight line can be found by calculating the change in 𝑦 over the change in 𝑥, we obtain that the slope of our tangent is equal to five minus negative five over 2.6 minus negative five, which is equal to 10 over 7.6, which is also equal to 1.316 to three decimal places.

Now that we’ve found an estimate for the slope of the tangent at the point 0.5, 𝑓 of 0.5, we can use this estimate for the slope to estimate the derivative of 𝑓 at 0.5 since the derivative is the slope function. We obtain that 𝑓 prime of 0.5 is approximately 1.316. Now, the answer you get for this question may differ, since it depends on how we draw out our tangent. And this is why this is only an estimate for the derivative of 𝑓 at 0.5, since it’s nearly impossible to draw the perfect tangent at this point. And therefore, the slope of our tangent will always be out from the actual derivative by a small amount. However, estimating the derivative using this method can be quite a useful tool, since we can still get roughly what the derivative should be without knowing the equation of 𝑓.

Now, we have covered a variety of examples of how we can estimate derivatives of functions, both numerically and graphically. Let’s recap some key points of the video.

Key Points

We can estimate the derivative of a function at a point numerically by finding the mean of the slope to the left and right of the point we’re trying to estimate the derivative. The formula is that 𝑓 prime of 𝑎 is approximately equal to 𝑓 of 𝑎 minus 𝑓 of 𝑏 over 𝑎 minus 𝑏 plus 𝑓 of 𝑐 minus 𝑓 of 𝑎 over 𝑐 minus 𝑎 all over two, where 𝑏 is less than 𝑎 and 𝑐 is greater than 𝑎. We can estimate the derivative of a function at a point graphically by estimating the tangent to the graph at that point and finding the slope of the tangent.