Differentiate
in Calculus Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

Question: differentiate ln [(1 - 3x^2) / (1+ 3x^2)] ^1/2

Here we have multiple applications of the chain rule, and one application of the quotient rule.

Let y = ln[w^(1/2)], where

w = u/v and u = (1 - 3x^2) and v = 1 + 3x^2. Then,

using du/dx = -6x and dv/dx = 6x,

dw/dx = (v.du/dx - u/dv/dx)/v^2 = [(1 + 3x^2).(-6x) - (1 - 3x^2).(6x)]/(1 + 3x^2)^2

dw/dx = [-6x - 18x^3 - 6x + 18x^3]/(1 + 3x^2)^2 = -12x/(1+3x^2)^2

dw/dx = -12x/(1+3x^2)^2

Continuing ...

y = ln[w^(1/2)] = (1/2)ln(w)

dy/dw = (1/2w)

and,

dy/dx = (dy/dw).(dw/dx) = (1/2w).(-12x/(1+3x^2)^2)

dy/dx = -6x/{[(1 - 3x^2) / (1+ 3x^2)] * (1 + 3x^2)^2}

dy/dx = -6x/{(1 - 3x^2)  * (1 + 3x^2)}

dy/dx = -6x/(1 - 9x^4)

by Level 11 User (80.8k points)

Related questions

1 answer
asked Nov 28, 2013 in Calculus Answers by anonymous | 184 views
1 answer
1 answer
asked Mar 25, 2015 in Calculus Answers by anonymous | 118 views
1 answer
asked Mar 25, 2015 in Calculus Answers by anonymous | 104 views
1 answer
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
83,155 questions
87,984 answers
1,978 comments
4,874 users