dy/dx If x=t^3 and y=ln(1-t) what is dy/dx

Expert Answers

An illustration of the letter 'A' in a speech bubbles

You need to find t in terms of x and then you should substitute the expression in terms of x for t in equation `y = ln(1 - t)`  such that:

`x = t^3 =gt x^(1/3) = (t^3)^(1/3) =gt root(3) x = t`

`y = ln(1 - root(3) x)`

You need to find `(dy)/(dx), ` hence you need to use chain rule such that:

`(dy)/(dx) = ((1 - root(3) x)')/(1 - root(3) x)`

`(dy)/(dx) = -((1/3)*x^(1/3 - 1))/(1 - root(3) x)`

`(dy)/(dx) = -((1/3)*x^(-2/3))/(1 - root(3) x)`

`(dy)/(dx) = -1/(3*x^(2/3)*(1 - root(3) x))`

`(dy)/(dx) = -1/(3*root(3)(x^2)*(1 - root(3) x))`

Hence, evaluating `(dy)/(dx)`  under given conditions yields `(dy)/(dx) = -1/(3*root(3)(x^2)*(1 - root(3) x)).`

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial