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:
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.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 (xk)' = kxk-1 we can use for any k.
(an unfinished article, to be continued).
