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Bir o’zgaruvchili funktsiyaning differentsial hisobi
7. Hosila


7.1. Hosilaning ta’rifi va differentsiallash qoidalari


Ta’rif. funktsiyaning hosilasi deb, funktsiya ortirmasi ∆y ni argumentning mos ortirmasi ∆x ga nisbatining ∆x→0 dagi limitiga aytiladi.
Berilgan funktsiyadan hosila topish amali shu funktsiyani differentsiallash deyiladi.
Uni quyidagicha belgilanadi:
, eki ,
funktsiyaning hosilasini hisoblash quyidagi to‘rt bosqichda bajariladi.
I. x, Δx ; x + Δx da: y + Δy = f(x + Δx)
II. Δy = f(x + Δx) – f(x)
III.
IV.


Differentsiallash qoidalari:
1.
> Diff(u+v, x)=diff(u(x)+v(x), x);

2. ,
> Diff(u*v, x)=diff(u(x)*v(x), x);

3.
> Diff(u*v*w, x)=diff(u(x)*v(x)*w(x), x);

> Diff(u(x)*v(x), x$3)=diff(u(x)*v(x), x$3);

4.
> Diff(u/v, x)=diff(u(x)/v(x), x);

5. y = f(U), U = φ(x) bo‘lsa,
> Diff(F(u(x)), x)=diff(F(u(x)), x);

6. ,
> Diff(u(x)^v(x), x)=diff(u(x)^v(x), x);

7. x=φ(t), y=ψ(t) ;
> x:=phi(t); y:=psi(t); dy/dx=diff(y,t)/diff(x,t);

> dy/dx =diff(si(t),t)/diff(hi (t),t);

> diff(F(x,y(x)), x)=0;

> d:=solve((D[1](F))(x,y(x))+(D[2](F))(x,y(x))*d, d);



HOSILALAR JADVALI


1. bo‘lganda,
2. hosilasi:
3. .
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7.1-misol. Quyidagi funktsiyalarning hosilasini toping.
1. 2.
3. 4.
5. 6.
Yechish.
1). - murakkab funktsiya bo‘lganligi uchun U(x)=1-sinx deb, y = ln(U(x)) funktsiya hosilasini topamiz. Murakkab funktsiya hosilasini topish qoidasiga asosan:
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