Derivation of a function

Derivative is a fundamental concept in differential calculus, it plays an important role in determining the progress of a function, for example, and is hated by students on the one hand, and on the other hand, the derivative can be calculated by a properly trained monkey. In this article, only the definition of derivative and related concepts will be described and explained. Solved examples can be found in the adjacent articles: simple examples on derivatives and more complex examples. The opposite process to derivation is integration.

What is derivation

Before we even get to the definition of derivative, let's talk about what we calculate by derivation in the first place, and what it can subsequently be good for.

By deriving a function, we get the directive of a tangent. That's probably a lot of dirty words together, so from the beginning. Simply put, a tangent line is a line that touches a given graph at exactly one point. It's not an exact definition, you can read that on Wikipedia for example, but it's roughly sufficient. Now take a look at the following figure:

The black curve is the graph of the function y = x². The blue line is the tangent to this function at the point D = [1,1], marked in red. The angle α, marked in green, is the angle that the tangent line makes with the axis x - more precisely, with the positive semi-axis x. Now let's define the notion of a directive tangent line. The tangent directive in this figure is the tangent of the angle alpha. The tangent is a classical goniometric function that allows us, for example, to calculate the angles and side sizes in a triangle. So the tangent directive is the tangent of the angle that the given tangent makes with the positive semi-axis x. And we get this directive just by derivation.

We also distinguish between the concepts of derivative of a function at a point and derivative of a function. The derivative of a function at a point is just the directive of the tangent at that point. The derivative of a function is then another function that prescribes a directive for a general argument x. An example follows.

Motivation

Using the derivative, we can thus compute the tangent directives. What can this be good for? Take a look at the following figure:

The figure again shows the function y = x² and the four tangents marked. Two green and two blue. Notice that the function y = x² is decreasing on the interval (−∞,0), while it is increasing on the interval (0,∞). But what also holds for their tangents, or the angle they make with the positive semi-axis? The blue tangents, which are the tangents that pass through the points belonging to the interval in which the function is increasing, make an angle of less than 90 degrees with the axis. While the green tangents make an angle with the axis that is greater than 90 degrees. How does this translate into the tangent directives?

For this we need to know the behavior of the tangent function. From the following graph we learn that if the angle has a magnitude of less than 90 degrees, then the value of the tangent is positive (the blue highlighted part); conversely, if the angle is greater than 90 degrees but less than 180 degrees (i.e., less than Pi radians), then the value of the tangent is negative. So what conclusion can we draw? If the directive (remember that the directive is just the tangent of the angle) of the tangent at a given point is positive, then the function at that point is increasing; if it is negative, then it is decreasing.

Definition

We have a long way to go before we get to the actual definition of the derivative. And we'll need the limit of the function to do that. If you don't know limits, go back to them or just skim the formulas, another chat probably won't be for you.

Before we move on to the tangent directive itself, we'll stop at the secant directive, which will be easier. The intercept is the line that intersects the graph at two points. Now let's try to derive how we would calculate this directive. Look at the picture:

We have the graph of the function x² + 1 and the intercept s, which intersects the graph at two points [0,5; 1,25] and [2, 5]. These points are highlighted in blue. Since we won't be too interested in specific values, these values are marked in the graph in general as a and b and f(a) and f(b). While f(a) is the value of the function f at the point x = a. Which is true: the value of our function x² + 1 at point x = 2 is 2² + 1 = 5.

Now it's a matter of deriving how to calculate the angle marked as alpha, i.e. the angle that the intercept makes with the axis x. We'll modify the figure a bit first to get some triangles in there, it will be easier to work with.

What have we done? We connected the points where the intersection line intersected with the graph of the function y = x² + 1 with a line segment and drew a right triangle ABC. What did we do to help ourselves? Because the angle labeled β is the same size as the angle α (they are congruent angles). But we know the sizes of the sides AB and BC. It is true that |AB| = b − a (it is simply the distance of the point b from a). Similarly, the length of |BC| = f(b)−f(a). How do we now calculate the size of the angle β, and α, respectively ?

He wants to know how to calculate the tangent. The tangent is the ratio of the opposite branch to the adjacent branch, so the tangent of angle alpha (beta) is equal to the ratio of the size of the side BC to the size of the side AB. Let's write it down:

$$\mbox{tan}(\beta)=\frac{|BC|}{|AB|}=\frac{f(b)-f(a)}{b-a}.$$

Since the directive is the tangent of the angle, we used this formula to calculate the directive of the intercept. But how does this help us when we want to calculate the tangent directive? Imagine that we bring the points A and C closer together until they merge. The approximation is shown in the figure:

The closer we bring the upper point of intersection to the lower point, the closer the line is to becoming a tangent. The original intersection/intercept CA was far from a tangent. If we bring the point C closer to the point A, we get, for example, the intercept C₁A, which is already more tangent-like. The slash C₂A is even more similar, etc.

When does a secant become a tangent? At the moment when the two points of intersection merge into one, i.e., when for some point C_n it would be true that C_n = A. At that moment we have made the intersection tangent, which is what we wanted. The question is how to calculate the directive, because we can't just subtract the two points because they are the same:

$$\mbox{tan}(\beta)=\frac{f(a)-f(a)}{a-a}=\frac00.$$

This won't work, we can't calculate it that way. It's not for nothing that we've only brought the points closer together. We will need a limit. Now we want to compute the tangent directive at a given point [a, f(a)] and we have the directives of all possible intercepts (we can already compute those). So we're going to gradually calculate the directives of the intercepts that we're going to approximate so that we get a tangent. So we will be approximating b to a. We cannot approximate the points so that they are equal, but we can approximate them so that their difference is boundedly close to zero. Vida, limita. So we will approximate b to a until they are limitingly equal. So we compute the limit when b approaches a. That's it, the formula looks like this:

$$\mbox{tan}(\alpha)=\lim_{b\rightarrow a}\frac{f(b)-f(a)}{b-a}.$$

We call this limit the derivative of the function f(x) at the point x = a. Recall that the angle α is equal to the angle β. Normally we do not use the variables a and b, but look for the derivative at the point x₀ and approximate with the point x. We can then write:

$$\mbox{tan}(\alpha)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}.$$

So far, as with the continuity function, we have defined the derivative at a single point. If we have a function f(x), which is derivable on the open interval I, then the values of this derivative define a function $f^\prime(x)$, which we call the derivative of the function.

We write the derivative using a superscript comma, like this:

$$(x^2)^\prime=2x,\qquad f^\prime(x)=,\ldots,\qquad f^\prime(x_0)=,\ldots$$

You may also encounter another form of notation, using dx:

$$\frac{d}{dx}f(x)=f^\prime(x)$$

So in the formula we used a moment ago, we can replace the tangent with the derivative notation:

$$f^\prime(x_0)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}.$$

We can also define a one-sided derivative from the right or left using one-sided limits. We won't go into that now, it's not that important.

A slightly different definition

Another definition of the derivative of a function at a point looks like this:

$$f^\prime(a)=\lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h}.$$

How do we arrive at such a definition? We just need to change the picture we started from a little bit. For we can express the distance of the point a from the point b as h = b − a. Thus we can write the point b as a + h, since h is just the distance of b from a. The picture would look like this:

At the bottom we have the points a and a + h, then f(a) and f(a + h) on the axis y. Now we make a triangle again and calculate the tangent of the angle beta. The only difference is that we have already calculated the length of the side AB, it is just h.

Proper and improper derivatives

We also have proper and improper derivatives, just like the limits of functions. What does this mean? A derivative is proper if the value of the derivative at a point is equal to some real number. A derivative is non-proprietary if it is equal to plus or minus infinity. How to visualize this geometrically? When is a directive "infinite"? Let's start with an easier question - what does a tangent whose directive is equal to some large number look like? The tangent graph (see above) shows that if the tangent value is high, then the angle is close to 90 degrees (if we are in the interval (0, 180)).

One could conclude that if the function at that point has a non-eigenderivative, then the tangent will be at right angles to the x axis, perpendicular to it. And it does. What might the graph of such a function look like, and at what point would the tangent be perpendicular to the x axis ? For example, the function $y=\sqrt[3]{x}$:

Basic Properties

• A function has a derivative at a point if the function is also defined in the epsilon neighborhood of that point. If this neighborhood did not exist, we would not reach the limit over which the derivative is defined.

• The uniqueness of the existence (or non-existence) of limits implies to us that if a derivative exists at a point, it is unique. No function has more than one derivative at a point. Either one or none.

• As with one-sided limits, if a function has a derivative at a point, then the derivatives on the left and right must be equal.

• Sometimes we need to know the so-called second derivative. This is nothing complicated, we simply derivative the function once and then derivative the result a second time. We mark this with two commas: f''(x) = (f'(x))'.

• An important property: if a function has a derivative at a point, then the function is continuous at that point. This is again based on the limit property. Note that this does not apply in reverse. If a function is continuous at a point, it does not mean that it is derivable there. A typical example is the function f(x) = |x|. The graph is to the vertex, you cannot compute the directive of the tangent at that vertex. You can try to calculate the derivative from left and right, they will be different.

  

• You can also use derivatives to solve the calculation of some limits, L'Hospital's rule is used for this.

References

See the adjacent article for a list of formulas for working with derivatives. Some more can be found on Wikipedia.

You can also see solved examples of derivations and then possibly harder examples on derivations. You can also find a lot of solved examples on the forum here.

If you need to quickly check the derivative calculation, you can use a math tool like Wolfram|Alpha. Just type "derive x^2+6x" (for example) in the input field there, and Wolfram will calculate the derivatives of the function you specify. You can also use the Czech MAW.

More material can be found, for example, on the CTU website.

In the literary world, you can read about derivation in the book The Second Derivative of Desire by Tomas Sedlacek. But it will probably be a different derivation :-).