Welcome back, everyone. So at this point in the course, we should be very familiar with how to find derivatives. So say we have a function in this form, y equals f of x. If we want to find the derivative f'(x), we know this is just taking the derivative of y with respect to x. Now, in this video, we're going to apply this concept to finding something called a differential.
You may hear the word differential and think it sounds like differentiation. Well, it turns out it is a similar idea because the differentials are just the individual pieces, dy and dx, that we've seen in this notation before. It turns out there are problems where we will be asked to find what these values are called differentials, and it there is a pretty straightforward process to do it if you're familiar with derivatives. So let's just get right into things. Now let's say we have an example where we're given the function f(x) = x³ + x, and we're asked to find the differential dy when x is equal to 2 and dx is equal to 0.1.
Now our mission ultimately is to find dy, and the way that I can do this is by thinking about how dy is related to the derivative of our function f of x. If I'm looking at this equation, this relationship I see right here, I could just take this dx and multiply it on both sides. This would be the strategy for isolating dy on one side of the equation. This is a way that you can think of finding the equation for the differential dy because it turns out the equation f'(x)dx is the relationship that you can use to find dy. So let's try it in this example.
First, I need to find f'(x). To do that, I'll use the power rule for this function we have. So using the power rule on x³, we're going to get 3x2. Then I'll add this to the derivative of x with respect to x, which is just 1. So this is my derivative f'(x).
Since I know that I need to multiply this whole thing by dx, that means we're going to have our derivative, 3x2 + 1, and that whole thing multiplied by dx. Notice all we really did here is took this derivative of our function y with respect to x, and we just moved this dx to the other side to get this function right down here. So this is really the strategy. From here, what I'm going to do is plug in the values that we have. I see that we have x is equal to 2 and dx is equal to 0.1.
We're going to have 3 times 22 + 1, and then that whole thing is going to be multiplied by 0.1. At this point, this is all math that you should be able to do by hand. And if you do this, you'll get this 22, which is 4 times 3, which is 12, plus 1 times 0.1, which all comes out to 1.3. So this is going to be the differential dy, and then that right there would be the solution to this example. So this is the strategy for finding these missing differentials, dy, in these types of situations.
But the question, of course, becomes, why would we need to know this? We've been spending some time recently talking about how these problems can apply to the real world, but how does any of this apply to the real world? Well, it turns out you can use this process for estimating or approximating values that are really hard to calculate by hand. So let's say we have this example down here where we are given this function and we are asked to estimate f of 2.1. Now notice, given the function we have, if I take 2.1 and replace x with it, I would have to use a calculator here, or it would be a very tedious by hand problem figuring out what 2.1 cubed is and then adding up all the decimal values.
But thankfully, we can use differentials to find this more simply. Now, as you can see, the example we have down here is actually the same numbers that we had up here. It's the same function and the same values. So what I'm going to do is I can say here that DX, we have as 0.1, the same as it was above. And then DY, I can see here is 1.3 based on what we calculated.
How can we use this information to approximate f of 2.1? Well, to do this estimation, what you need to realize is what the dy and the dx really are if we think about a graph. Here we have some function, which could be any function that we have. And if we draw a line tangent to this function at a certain point, notice how this delta x here is going to be the change in your x position on the graph, and delta x is going to be the same thing as dx.
These two values are going to be the same. Now dy in this case is going to be this change on the y axis when we're looking at this line. And notice how dy is very close to the same thing as delta y, but it's not quite the same. So, you can think of this dy and dx as being a rise over a run. And we know the derivative is just the slope of the tangent line.
So if you look here, you can see that the change in x is going to be the same as dx, and the change in y is going to be close to dy. What that means is we can approximate this to be a similar value. So we can see that f of x + our differential dx is going to be approximately equal to our function value f of x + dy, this little portion right here. So what I'm going to say is that f of x + dx is approximately equal to f of x + the value dy. We have f of x, which I'm going to say x is 2 in this case since we have an x value of 2, plus our differential dx, which is 0.1.
And notice here that that's going to estimate f of 2.1 because that's the same thing. And this whole thing will approximately be equal to our function f of 2 with our 2 value plugged in, plus our differential dy, which I can see is 1.3. Now f of 2, I can calculate by just plugging 2 into this equation up here. So we're going to have 2 cubed plus 2, which this whole thing will come out to 10. So we're going to end up getting is 10 + 1.3, and this whole thing is just equal to 11.3.
This right here would be the solution to our problem, which is the approximate value of f of 2.1. Now you could actually take f of 2.1 and put it into your equation. You would just likely have to use a calculator to get your answer. But I think you would find the answer you get on a calculator is very similar to this answer right here. It might be slightly off with the decimal approximation, but it's going to be very close.
And that's really the idea of using differentials to estimate certain values. So say that you were a scientist and there was a really complicated equation you had, you could just use derivatives to find the individual differentials and solve. You just have to take whatever number that you're looking at and split it into 2 values that are relatively simple to calculate. So this is the process for differentials. I hope you found this video helpful, and let's try getting some more practice.