Discussion
If \(x^{y} y^{x} = C\), then \(\frac{dy}{dx} = ?\)
Answer: A
Take natural logarithms of \(x^y y^x = C\) to get \(y\ln x + x\ln y = \ln C\) and differentiate implicitly:
\[ \begin{aligned} \frac{y}{x} + \ln x\,y' + \ln y + \frac{x}{y}y' = 0 \\ y' = -\frac{(y + x\ln y)/x}{(y\ln x + x)/y} \\ &= -\frac{y(y + x\ln y)}{x(x + y\ln x)} \end{aligned} \]The negative sign is essential because the two sides of the constraint trade off against each other. The unsigned version in the second option and the options with x and y swapped (the third and fourth) come from errors in collecting the terms with y' or in inverting the fraction.
No comment is present. Be the first to comment.
Loading…
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development