Introduction to derivatives

For many practical and scientific tasks, people have to analyze functions to find maxima and minima, intervals of increase and decrease. To perform such analysis, derivatives of the functions are used. For example, it is applied to find the velocity in physics or the optimal volume of production in economics.

In this topic, we will study the concept of derivatives and prepare for further immersion in differentiation. To begin with, let's take a look at the problem that leads to the notion of a derivative.

Behaviour of function

Imagine a straight road passing through a hilly terrain: it goes up and down. If the $Ox$ axis is directed horizontally along the road, and $Oy$ is directed vertically, then the road line will be very similar to the graph of some continuous function:

Moving forward along such a road, we also move up or down. We can also say: when the argument changes (movement along the $Ox$ axis), the value of the function changes (movement along the $Oy$ axis).

How to determine the "slope" of our road, what kind of value it can be? This is the next value: how much the height will change when moving forward a certain distance. We will denote the argument changes by $\triangle x$ (read "delta $x$"). That is, $\triangle x$ is a change in the value of $x$, $\triangle y$ is a change of $y$; then $\triangle f$ is the change of $f$.

So, we have moved forward horizontally by $\triangle x$. If we compare the road line with the graph of the function $f(x)$, then we denoted the rise by $\triangle f$. That is, when moving forward by $\triangle x$, we rise higher by $\triangle f$.

The value of $\triangle f$ is easy to calculate: if at the beginning we were at the height $f_1$, and after moving we were at the height $f_2$, then $\triangle f = f_2-f_1$. If the end point is lower than the start point, $\triangle f$ will be negative – this means that we are not going up, but going down.

The value of "slope" indicates how much (steep) the height increases as you move forward one unit of distance:

$$\dfrac{\triangle f}{\triangle x}$$For example, if the road goes down 0.6 km while advancing 200 m? Then the slope is equal to $$\dfrac{\triangle f}{\triangle x} = \dfrac{-600}{200} = -3$$Note that if we look at the top of a hill and take the beginning of the section half a kilometer before the top, and the end half a kilometer after it, it can be seen that the height is practically the same:

It turns out that the slope here is almost zero, which is clearly not true. This is due to the fact that a lot can change at a distance of 1 km and it is necessary to consider smaller sections for a more adequate and accurate assessment of the slope. For example, if you measure the change in height when you move one meter, the result will be much more accurate. But even this accuracy may not be enough for us. What distance then to choose? Less is better. For this, the concept of infinitely small is used, that is, the value is in absolute value less than any number that we can only name.

If we want to write that the value $x$ is infinitesimal, we write $x \to 0$ (we read "$x$ tends to zero"). Note that this number is not zero, but very close to it. This means that you can divide by it.

We can conclude that a well-calculated slope is the slope, calculated for an infinitesimal distance, that is:

$$\dfrac{\triangle f}{\triangle x}\ \ x \xrightarrow[\triangle x \to 0]{} 0$$

The concept of the derivative

The derivative of a function is the ratio of the increment of the function to the increment of the argument at an infinitely small increment of the argument.

How much the argument $x$ has changed as it moves along the $Ox$ axis is called the argument increment and is denoted by $\triangle x$.

How much the function $f(x)$ has changed while moving forward along the $Ox$ axis by a distance $\triangle x$ is called the function increment and is denoted by $\triangle f$. So, the derivative of a function is

$$\dfrac{\triangle f}{\triangle x} \ \ \triangle x \to 0$$The derivative of a function $f(x)$ is denoted by $f'(x)$ or $\dfrac{df}{dx}$. Thus,

$$f'(x) = \lim\limits_{\triangle x \to 0} \dfrac{\triangle f}{\triangle x}$$

More about increments

So, we change the argument by the value $\triangle x$. What value do we change it from and what is the argument equal to?

Consider a point with coordinate $x_0$. The value of the function in it is $f(x_0)$. Then we make that increment: we increase the $x_0$ coordinate by $\triangle x$. Then the argument will be $x_0+\triangle x$, and the value of the function will now be equal to $f(x_0+\triangle x)$.

What about the function increment? This is still the amount by which the function has changed:

$$\triangle f = f(x_0 + \triangle x)-f(x_0)$$For example, let's find the increment of the function $f(x) = 5x+8$ at the point $x_0$ with the argument increment equal to $\triangle x$:

$$\triangle f=f(x_0 + \triangle x)-f(x_0) = 5(x_0 + \triangle x) + 8-(5x_0+8) = \\ = 5x_0 + 5\triangle x + 8-5x_0-8 = 5\triangle x$$Now we can find the derivative of the function $f(x) = 5x+8$:

$$f'(x) = (5x+8)' = \lim\limits_{\triangle x \to 0} \dfrac{\triangle f}{\triangle x} = \lim\limits_{\triangle x \to 0} \dfrac{5\triangle x}{\triangle x} = 5$$So, the derivative of the function $f(x) = 5x+8$ is equal to $5$.

Conclusion

In this topic, we have studied the concept of derivative and learned the principle of its finding.

• The derivative of a function is the ratio of the increment of the function to the increment of the argument at an infinitely small increment of the argument.
• Derivative is the limit of this ratio when the argument increment tends to $0$.
• The negative derivative indicates the decrease of the function, the positive one shows its increase.

In practice the derivative does not have to be calculated from scratch every time. We will study it in the following topics how to simplify this process!