Derivative of a Function.

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 = ax³, where a is any constant..



We've used Newton's Formula:



3. y = xⁿ, n - natural number.



We've used Newton's Formula again.

The same formula for derivative (x^k)' = kx^k-1 we can use for any k.


(an unfinished article, to be continued).