Question

Differentiate following with respect to x using logaritms.
25) (2-x^3)^1/3 / x

Answer 1

\(y=\frac{(2-x^3)^\frac{1}{3}}{x}\\\ln(y)=\ln(\frac{(2-x^3)^\frac{1}{3}}{x})=\frac{1}{3}\ln(2-x^3)-\ln(x)\)

Answer 2

you with me?

Answer 3

Srry. OS lag..

Answer 4

now we can differentiate term by term, and use explicit differentiation

\(\frac{1}{y}\frac{dy}{dx}=\frac{1}{3}\frac{-3x^2}{2-x^3}-\frac{1}{x}\\\frac{dy}{dx}=y(\frac{1}{3}\frac{-3x^2}{2-x^3}-\frac{1}{x})\)

now sub in for \(y\)

\(\frac{dy}{dx}=(\frac{(2-x^3)^\frac{1}{3}}{x})(\frac{1}{3}\frac{-3x^2}{2-x^3}-\frac{1}{x})\)

Answer 5

can you simplify?

Answer 6

will try

Answer 7

For the 2nd part..
like this ?