Question

segue a pergunta abaixo

Answer 1

a)

y=[(-x+2)/(3x+1)]^4

y'=4*[(-x+2)/(3x+1)]³ *[(-x+2)/(3x+1)]'

[(-x+2)/(3x+1)]' =[(-x+2)'*(3x+1)-(-x+2)*(3x+1)']/(3x+1)²

[(-x+2)/(3x+1)]' =[(-1)*(3x+1)-(-x+2)*(3)]/(3x+1)²

[(-x+2)/(3x+1)]' =[-3x-1+3x-6]/(3x+1)²

[(-x+2)/(3x+1)]' =-7/(3x+1)²

y'=4*[(-x+2)/(3x+1)]³ *(-7/(3x+1)²)

y'=-28*[(-x+2)/(3x+1)]³ *(1/(3x+1)²)

b)

y'=[(-2x²+1)²]'*(4+x)² + (-2x²+1)² [(4+x)²]'

y'=[2(-2x²+1)*(-4x)]*(4+x)² + (-2x²+1)² [2(4+x)*1

c)

y=sen(x²-4)

y'=(x²-4)' * cos(x²-4)

y'=2x*cos(x²-4)

d)

y=-cotg(3x²)

u=3x²

y=-cotg(u)

y'=-(3x²)' * [cotg(u) ]'

y'=-(6x) * [cos(u)/sen(u)]'

y'=-(6x) * [-sen(u)*sen(u) -cos(u)*cos(u)]/sen²(u)

y'=-(6x) * [-sen²(u) -cos²(u)]/sen²(u)

y'=-(6x) * [-1]/sen²(u)

y'=6x/sen²(u)

y'=6x/sen²(3x²)=6x * cossec²(3x²)