Question

Please provide an explanation and steps

Answer 1

Haven't done one of these in forever. We avoid the quotient rule with an implicit logarithmic differentiation.

`y=((x^2+x+1)(x^3+2x+5))/((2x-1)^5)`

`\ln y = \ln(x^2+x+1) + \ln(x^3+2x+5) - 5\ln(2x-1)`

Taking derivatives with respect to x,

`y'/y = (2x+1)/(x^2+x+1) + (3x^2+2)/(x^3+2x+5) - (10)/(2x-1)`

`y' = y \left( (2x+1)/(x^2+x+1) + (3x^2+2)/(x^3+2x+5) - (10)/(2x-1) \right)`

`y' = ((x^2+x+1)(x^3+2x+5)(2x-1))/((2x-1)^6) \left( (2x+1)/(x^2+x+1) + (3x^2+2)/(x^3+2x+5) - (10)/(2x-1) \right)`

`y'(2x-1)^6 =(2x+1)(x^3+2x+5)(2x-1)+(3x^2+2)(x^2+x+1)(2x-1) \\\quad -10(x^2+x+1)(x^3+2x+5)`

`y'=(-7 x^4 - 16 x^3 - 51 x^2 - 70 x - 57)/((2x-1)^6)`