Discussion
If \(y = \sin x + e^{x}\), then \(\frac{d^{2}x}{dy^{2}}\) is equal to
Answer: E
\(\frac{d^{2}x}{dy^{2}} = \frac{\sin x - e^{x}}{(\cos x + e^{x})^{3}}\).
Here \(\frac{dy}{dx} = \cos x + e^{x}\) and \(\frac{d^{2}y}{dx^{2}} = -\sin x + e^{x}\). For the inverse function,
\[ \begin{aligned} \frac{dx}{dy} &= \frac{1}{y'} \\ \frac{d^{2}x}{dy^{2}} &= \frac{d}{dx}\left(\frac{1}{y'}\right)\frac{dx}{dy} = -\frac{y''}{(y')^{3}} \end{aligned} \]
\[ \begin{aligned} \frac{d^{2}x}{dy^{2}} &= -\frac{e^{x} - \sin x}{(\cos x + e^{x})^{3}} \\ &= \frac{\sin x - e^{x}}{(\cos x + e^{x})^{3}} \end{aligned} \]
The options with a square in the denominator forget the chain-rule factor \(\frac{dx}{dy}\), and \(\frac{e^{x} - \sin x}{(\cos x + e^{x})^{3}}\) drops the minus sign.
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