**Definition & Notation.**

def. Derivative of a Function

*y = f(x)*in point

*x*is limit to which closes the ratio of increment of a function

*Δy*to an increment of a variable

*Δx*, when increment of a variable

*Δx*closes to zero.

if such limit does not exist, then function has no derivative in this point.

derivative of a function

*y = f(x)*we can note as:

**Geometrical interpretation.**

Geometrically, a derivative of a function is equal to a tanget of angle

*α*, between the tangent line touching the graph of the function in a point

*x*and positive direction of

the OX axis.

**Additional notes.**

Finding a derivative of a function we can call 'function differentiation'.

From definition we can see that derivative of a function is a quickness of the change of a function

*f(x)*, when

*x*changes.

**Calculations.**

Let's calculate a few of function derivatives, using definiton.

1.

*y = sin(x)*.

We've used one of trigonometrical formulas (last one, for the sines difference):

... as well, as that is:

Similarly:

*(cos x)' = - sin x;*

2.

*y = ax*.

^{3}, where*a*is any constant.We've used Newton's Formula:

3.

*y = x*

^{n}, n - natural number.We've used Newton's Formula again.

The same formula for derivative

*(x*we can use for any k.

^{k})' = kx^{k-1}(an unfinished article, to be continued).

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