y=f(x), . x y=f(x) .
: x 0 x. x x 0 x x 0 Δx. , Δx = x x 0 ( , ). , x=x 0 +Δx, .. . , x 0 f(x 0 ), x f(x) = f(x 0 +Δx).
y y 0 = f(x) f(x 0 ) y = f(x) x 0 Δy. ,
Δy = f(x) f(x 0 ) = f(x 0 +Δx) - f(x 0 ). | (1) |
x 0 , x . y 0 = f(x 0 ) , y = f(x) . Δy Δx (1) , Dy Δx.
Δx →0. , f(x) x 0 f '(x 0). ,
.
y = f(x) x 0 Δ y Δ x, .
, x , .. x. f '(x)
f ' (x),y ', . x = a f '(a) y ' |x=a.
f(x) .
:
- x Δ x f(x + Δx).
- Δy = f(x + Δx) f(x).
- Δ x ∞0.
. , f (x) x = a,
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. . x = − 1 x = 1. , x = 1. , . x = − 1 , . |
- ( . arc ).
, ( , ), .
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y = arcsin x y = sin x. -, , , y = arcsin x . , \left [ -\frac{\pi}{2}; \frac{\pi}{2} \right ]. y = sin x \left [ -\frac{\pi}{2}; \frac{\pi}{2} \right ] , y = arcsin x, y = sin x \left [ -\frac{\pi}{2}; \frac{\pi}{2} \right ] y = x. | |
y = arccos x y = cos x. -, , , y = arccos x . , [0;\pi]. y = cos x , , [0;\pi] y =arccosx, y = cos x [0;\pi] y = x. | |
y = arctg x y = tg x. -, , , y = arctgx . , \left(-\frac{\pi}{2}; \frac{\pi}{2} \right). y = tg x , , \left(-\frac{\pi}{2}; \frac{\pi}{2} \right) y = arctg x, y = tgx \left(-\frac{\pi}{2}; \frac{\pi}{2} \right) y = x. | |
y = arcctg x y = ctg x. -, , , y = arcctg x . , (0;\pi). y = ctg x , , (0;\pi) y = arcctg x, y = ctg x (0;\pi) y = x. |
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