Question

Use logarithmic differentiation to find the derivative of y. y = (x3 + 1)5(x - 1)ºx2 (x3 + 1)(x - 1)3x2 (x2 + 115(x - 1)3x2 153 x3 + 1 + X-1 2 O 15x2 x3+1 3 + X-1 х O (x3 + 1) =(x - 1)3x2(51n(x3 + 1) + 3in(x - 1) + 2ln x)

Answer 1

first apply natural logarithm to both sides

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

move the exponents

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

find the derivative for each term

`\begin{gathered} (y^(\prime))/(y)=5(3x^2)/(x^3+1)+3(1)/(x-1)+2(1)/(x) \\ \\ (y^(\prime))/(y)=(15x^2)/(x^3+1)+(3)/(x-1)+(2)/(x) \end{gathered}`

multiply all the terms by y

`y^(\prime)=((15x^2)/(x^3+1)+(3)/(x-1)+(2)/(x))y`

replace y from the initial equation

`y^(\prime)=((15x^2)/(x^3+1)+(3)/(x-1)+(2)/(x))((x^3+1)^5(x-1)^3x^2)`

so the solution is the second option