Во всех примерах используется таблица производных
1. y=1/x
y' = (1/x)' =(x^(-1))' =(-1)*x^(-2)= -1/x^2
2. y =(1/x^(1/3)
y'= (x^(-1/3))' = -1/3*x^(-4/3)=-1/3*∛x^4
3. y = ctgx/4
y'= 1/4(ctgx)'=1/4(-1/(sin^2(x)=-1/4sin^2(x)
4. y= lnx/(3x^2+4)
используем формулу частного: (u/v)' = (u'v-uv')/v^2 (1)
y'= (lnx/(3x^2+4))' =((lnx)'*(3x^2+4)- lnx*(3x^2+4)')/(3x^2+4)^2 =
=(1/x*(3x^2+4)- lnx*6x)/(3x^2+4)^2 = 3x^2+4-6*x^2*lnx /x*(3x^2+4)^2
5. y=√x
y'=(√x)' = (x^1/2)' =1/2*x^(-1/2) = 1/2√x
6, y=1/x^2 - 3*1/√x+8
y'=(1/x^2 - 3*1/√x +8)' =(x^(-2) -3*x^(-1/2) +8)' = -2*x^(-3)-3*x^(-3/2) =
= -2/x^(3) + 3/√x^3
7. y= (x+5)^7 +√2
y'= ((x+5)^7 +√2)' = 7*(x+5)^6*(x+5)' = 7*(x+5)^6
8. y=xlnx
используем формулу: (uv)' = u'v+uv'
y'=(x*lnx)' = (x)'*lnx+x*(lnx)' = lnx + x*1/x = lnx +1
9. y = (2-3x)^5
y' = ((2-3x)^5)' = 5*(2-3x)^4*(2-3x)' =5*(2-3x)^4*(-3) =-15*(2-3x)^4