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## The Derivative of a Function at a Point - Mathematics LibreTexts

Learn how. Derivative [ n 1 , n 2 , … ] [ f ] is the general form, representing a function obtained from f by differentiating n 1 times with respect to the first argument, n 2 times with respect to the second argument, and so on. So this is the. This is the.

## 1.3: The Derivative of a Function at a Point

This is the point five This is This is , actually, let me write out the this is one, two, three, four, five, six, seven, eight, nine, and ten. Now they want us to know, they want to what is the derivative of our function when F is equal to four? Well, they haven't told us even what the value of F is at four.

We don't know what that point is.

### Tangent and Normal Lines

But what they're trying to do is, well, we're trying to make a best estimate. And using these points, we don't even know exactly what the curve looks like. It could look like all sorts of things.

We could try to fit a reasonably smooth curve. The curve might look something like that. But it might be wackier.

## Find the derivative of the function and evaluate the derivative at the given value of a.

It might do something like this. Let me try to do it. It might look something like this. So we don't know, for sure. All we know is that it needs to go through those points.

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But let's just, for the sake of this exercise, let's assume the simplest, let's say it's a nice smooth curve without too many twists and turns that goes through these points, just like that. So what they're asking, okay, when X is equal to four if this yellow curve were the actual curve then what is the slope of the tangent line, at that point? So we would be visualizing that. Now to be clear, this tangent line that I just drew this would be for this version of our function that I did connecting these points.

That does not have to be the actual function. We know that the actual function has to go through those points. But I'm just doing this for visualization purposes. One of the whole ideas here is that all we do have is the sample and we're trying to get a best estimate.

We don't know if it's even gonna be a good estimate. It's just going to be a best estimate.

### How to find the derivative of the function at some value of x?

So what we generally do when we just have some data around a point, is, let's use a data points that are closest to that point and find slopes of secant lines pretty close around that point. Determining the lines tangent to the graph of a function from a point outside the function, example. Note that in the above equation x is the given fixed point and the only variable quantity is the distance.

As the distance D x gets smaller, the average rate of change becomes more accurate, that is the secant line becomes more and more like the tangent line of the curve at the point x , f x , as shows the figure above. To find the coordinates x 0 and y 0 of the points of tangency we should solve the following two equations,.

The first equation states that the point of tangency must lie on the given function while, at the same time, its coordinates x 0 and y 0 must satisfy the equation of the tangent line 2 , since the point of tangency is the common point of the tangent and the curve. Then substitute the slope and the y 0 into second equation to get the points of tangency,.