Task/25860011
---------------------
1.
------------------------
а) y =x⁴ , x₀ = -1.
y ' =(x⁴) ' = 4x³ .
y '(x₀) =y '(-1) = 4*(-1)³ = - 4.
---
б) y =sinx -cosx , x₀ = π/4 .
y ' =(sinx -cosx) ' =(sinx) ' - (cosx) ' =cosx -(-sinx) =cosx +sinx .
y '(x₀) =y ' (π/4) =cos(π/4) +sin(π/4) =(√2)/2 +(√2)/2 =√2.
---
в) y = -3cosx+2sinx , x₀ = π/4 .
y ' = (-3cosx+2sinx ) ' =(-3cosx) ' +(2sinx ) ' = -3cosx) ' +2(sinx ) ' =3sinx +2cosx.
y '(x₀) =y ' (π/4) =3sin(π/4) +2cos(π/4) =3*(√2)/2 +2(√2)/2 =2,5√2.
---
г) y = 1 +∛x , x₀ = 2√2.
y ' = (1 +∛x) ' =1 ' + (∛x) ' =0 +(1/3)*x ^(-2/3)= (1/3)*(1 /x ^(2/3) )
y ' = (1/3∛x²)
y (x₀)= y ' (2√2) = (1/3∛(2√2)² =1/3∛(2²*(√2)² )= 1/3∛8 = 1/(3*2) = 1/6.
2.
------------------------
а) y =4x³ *x⁴ =4x⁷ ; б) y = 1/x⁴ = x⁻⁴ ;
y ' =4*7*x⁷⁻¹ =28x⁶ . y ' = - 4*x⁻⁴⁻1 = - 4x⁻⁵ .
------ ------
в) y =1/(7x⁶) =(1/7)*x⁻⁶ ; г) y =x⁴/124 ;
y ' = (1/7)*x⁻⁶ ⁻¹ = (1/7)*x⁻⁷. y ' =(1/124)*(x⁴) ' =(1/124)*4*x³ =(1/31)x³ .