13 uždavinys

Apskaičiuokite funkcijos $$(x-3)^{2}-6\cdot x^{2}$$ išvestinę.

Sprendimas.

 ( (x-3)^²- 6* x^²)′  = 

 ( (x-3)^²- 6* x^²)′ = $$((x-3)^{2}-6\cdot x^{2})'$$ = 

$${\normalsize ((x-3)^{2}-6\cdot x^{2})'}$$ = $${\normalsize ((x-3)^{2})'-(6\cdot x^{2})'}$$

Paaiškinimas:

Sumos išvestinė (f+g)′ = f′ + g′

 ( (x-3)^²)′- ( 6* x^²)′ = $$((x-3)^{2})'-(6\cdot x^{2})'$$ = 

$${\normalsize (x-3)^{2}}$$ = $${\normalsize x^{2}-2\cdot x\cdot 3+3^{2}}$$

Paaiškinimas:

Pagal skirtumo kvadrato formulę
$${\normalsize (a-b)^{2} = a^{2}-2\cdot a\cdot b+b^{2}}$$
a = x, b = 3

 ( x^²- 2* x* 3+ 3^²)′- ( 6* x^²)′ = $$(x^{2}-2\cdot x\cdot 3+3^{2})'-(6\cdot x^{2})'$$ = 

$${\normalsize 2\cdot x\cdot 3}$$ = $${\normalsize 6\cdot x}$$

 ( x^²- 6* x+ 3^²)′- ( 6* x^²)′ = $$(x^{2}-6\cdot x+3^{2})'-(6\cdot x^{2})'$$ = 

$${\normalsize (x^{2}-6\cdot x+3^{2})'}$$ = $${\normalsize (x^{2})'-(6\cdot x)'+(3^{2})'}$$

Paaiškinimas:

Sumos išvestinė (f+g)′ = f′ + g′

 ( x^²)′- ( 6* x)′+ ( 3^²)′- ( 6* x^²)′ = $$(x^{2})'-(6\cdot x)'+(3^{2})'-(6\cdot x^{2})'$$ = 

$${\normalsize (x^{2})'}$$ = $${\normalsize 2\cdot x}$$

Paaiškinimas:

Laipsnio išvestinė, kur n = 2

 2* x- ( 6* x)′+ ( 3^²)′- ( 6* x^²)′ = $$2\cdot x-(6\cdot x)'+(3^{2})'-(6\cdot x^{2})'$$ = 

$${\normalsize (6\cdot x)'}$$ = $${\normalsize 6}$$

Paaiškinimas:

x išvestinė yra 1

 2* x-6+ ( 3^²)′- ( 6* x^²)′ = $$2\cdot x-6+(3^{2})'-(6\cdot x^{2})'$$ = 

$${\normalsize (3^{2})'}$$ = $$0$$

Paaiškinimas:

Konstantos išvestinė yra 0

 2* x-6+0- ( 6* x^²)′ = $$2\cdot x-6+0-(6\cdot x^{2})'$$ = 

$${\normalsize (6\cdot x^{2})'}$$ = $${\normalsize 12\cdot x}$$

Paaiškinimas:

Laipsnio išvestinė, kur n = 2

 2* x-6+0- 12* x = $$2\cdot x-6+0-12\cdot x$$ = 

 2* x-6- 12* x = $$2\cdot x-6-12\cdot x$$ = 

 2* x- 12* x-6 = $$2\cdot x-12\cdot x-6$$ = 

$${\normalsize 2\cdot x-12\cdot x}$$ = $${\normalsize -10\cdot x}$$

- 10* x-6$$-10\cdot x-6$$

$$((x-3)^{2}-6\cdot x^{2})'$$  = $$$$

$$(x^{2}-2\cdot x\cdot 3+3^{2})'-(6\cdot x^{2})'$$  = $$$$

$$(x^{2})'-(6\cdot x)'+(3^{2})'-(6\cdot x^{2})'$$  = $$$$

$$2\cdot x-6+0-12\cdot x$$  = $$$$

$$-10\cdot x-6$$ $$$$

((x-3)^2-6*x^2)′  = 

(x^2-2*x*3+3^2)′-(6*x^2)′  = 

(x^2)′-(6*x)′+(3^2)′-(6*x^2)′  = 

2*x-6+0-12*x  = 

-10*x-6

Atsakymas: -10*x-6