Differentiation Using Limits

Posted September 1, 2014

by Lensyl Urbano

We can use the idea of limits to come up with some general relationships between functions and their slopes. Take, for example, the last project where we found the slope of the function y = x2 at the point where x = 3:

The green line is the curve y = x2 and the straight red line is the tangent to the curve at the point where x = 3 (i.e. at (3,9)).

The green line is the curve y = x2 and the straight red line is the tangent to the curve at the point where x = 3 (i.e. at (3,9)).

Finding the approximate slope using a forward difference.

Finding the approximate slope using a forward difference.

We found the slope of the tangent line at the point (3,9) by a series of approximations. First we took two points on the curve, (3,9) and (4,16) and found the slope between those two points.

The equation for slope can be written in any of these three ways:

slope-line

we find the exact slope by taking points on the line closer and closer together (which means that Δx is getting smaller and smaller). In math-speak, we’re saying that we’re taking the limit of the slope equation as Δx approaches zero.

slope-limit

Since we’re taking Δx to zero we might as well ask what happens to the slope when Δx is equal to zero.

slope-undef

As we can see from the equation, we end up with zero on the denominator, which makes the whole thing undefined, which we really do not want.

But what if we can rearrange things to get the Δx out of the denominator?

Let’s rewrite the equation y = x2 as a function:

f(x) = x^2

So the point we’re interested in find the slope at is just (x, f(x)), which in this case happens to be (3, 9), but we’re not going to be using the actual numbers anymore so we can come up with a more general relationship.

To find the slope we find the point where the value on the x-axis is x and the value on the y-axis is f(x).

To find the slope we find the point where the value on the x-axis is x and the value on the y-axis is f(x).

Now, and this is often the tricky part, the second point we use is going to have an x value of x + Δx:

For the second point, the x value we use is the first x offset by Δx.

For the second point, the x value we use is the first x offset by Δx.

which means that the value on the curve is f(x+Δx):

Our second point is where the x value is x+Δx and the y value is f(x+Δx).

Our second point is where the x value is x+Δx and the y value is f(x+Δx).

So lets carefully observe the notation here. To find the slope of a line we can use the equation:

slope-basic
But with the function notation:

  • y1 = f(x)
  • y2 = f(x+Δx)

so:

slope-f

Now watch very carefully as I replace the function notation with the actual functions, specifically:

  • f(x) = x2
  • f(x+Δx) = (x+Δx)2

to give:

slope-f2

and if you understand how this, we’re almost all there, because the rest is algebra.

We simplify the equation above by expanding the numerator.

slope-f3

now we can subtract the similar terms (x2) and divide through by Δx to get:

slope-f4

But remember we don’t just want the slope, we want to find the slope where Δx approaches zero:

slope-lim2

The problem before was that if we made Δx = 0 the equation would be undefined. But now, however, as Δx = 0 the second term in the equation just goes to zero:

slope-f5

leaving us just with the first term 2x:

slope-diff

Remember that we were trying to do this at the point where x = 3. So if we put x = 3 into this equation we get:

slope-diff2

Which we know is the right answer because we did the very problem by hand already.

but now however we’ve come up with a more general equation for the slope. With it we can easily find the slope of our curve at any point along the curve!

For example, what is the slope of the curve when x = 0:

slope-diff4

Notation

Now it’s a bit cumbersome to write the limit as Δx goes to zero every time, so we’ll instead call our equation for the slope of the line the differential, and we’ll give it the notation as the function prime:

i.e. if we have a function f(x) = x2:

diff_notation-1

we write its differential as:

diff-notation-2

To confuse things (at least for the moment) there are a number of ways of writing the differential (the different methods are useful in different contexts), so you will see things like:

diff-notation-3

So now that we know how to find the differential using limits, we’ll practice finding the differential of polynomial functions and see if we can find a general pattern that allows us to bypass the whole limits thing altogether.

Citing this post: Urbano, L., 2014. Differentiation Using Limits, Retrieved July 29th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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