One of the main subjects in the school program is differentiation or, speaking to more clear language, a function derivative. Usually it is difficult to pupil to understand what is a derivative and what its physical sense. The answer to this question can be received if to penetrate into the physical and geometrical meaning of a derivative. In this case, the lifeless formulation gains obvious sense even for the humanist.
In any textbook you will meet definition that the derivative is the relation of increment of function to increment of an argument, on condition of the increment of an argument tending to zero. Speaking to more clear and simple language, the word increment can be replaced safely with the term change. The concept of aspiration to zero argument should be explained to the pupil after passing of the concept "limit". However, most often these formulations meet much earlier. For understanding of the term "tends to zero" it is necessary to imagine it is insignificant the small size which is so small that it cannot be written down mathematically.
Similar definition seems to the pupil tangled. For simplification of a formulation, it is necessary to penetrate into the physical meaning of a derivative. Remember any physical process. For example, movement of the car on the section of the road. From a school course of physics it is known that the speed of this car is the relation of the passable distance at the right time for which it is passed. But in this way it is impossible to determine the instantaneous velocity of the car in concrete timepoint. When performing division the average speed on all the site of a way turns out. The fact that somewhere the car stood on the traffic light, and somewhere went under the hill with greater speed is not considered.
The derivative allows to solve this complex problem. Function of the movement of the car is presented in the form infinitesimal (or short) time intervals, on each of which it is possible to apply differentiation and to learn function change. For this reason, in definition of a derivative there is a mention about infinitesimal increment of an argument. Thus, the physical meaning of a derivative is that it is function change speed. Having differentiated function of speed on time it is possible to receive value of speed of the car in concrete timepoint. This understanding is useful when studying any process. In the surrounding real world there are no ideal correct dependences.
If to speak about the geometrical meaning of a derivative, then it is enough to imagine the schedule of any function which is not rectilinear dependence. For example, branch of a parabola or any wrong curve. To this curve it is always possible to carry out a tangent, and the common ground of a tangent and the schedule and will be required value of function in a point. The corner under which this tangent to abscissa axis is carried out and is defined by a derivative. Thus, the geometrical meaning of a derivative is a tilt angle of a tangent to a function graph.