MHT CET · Maths · Differentiation
If \(x^{\mathrm{y}}+\mathrm{y}^x=\mathrm{a}^{\mathrm{b}}\), then \(\frac{\mathrm{dy}}{\mathrm{d} x}\) at \(x=1, \mathrm{y}=2\) is
- A \(-2(1+\log 2)\)
- B \(2(1+\log 2)\)
- C \(2+\log 2\)
- D \(1+\log 2\)
Answer & Solution
Correct Answer
(A) \(-2(1+\log 2)\)
Step-by-step Solution
Detailed explanation
\(\frac{\mathrm{d}}{\mathrm{d}x}(x^{\mathrm{y}}) + \frac{\mathrm{d}}{\mathrm{d}x}(\mathrm{y}^x) = \frac{\mathrm{d}}{\mathrm{d}x}(\mathrm{a}^{\mathrm{b}})\) \(x^{\mathrm{y}}\left(\frac{\mathrm{dy}}{\mathrm{d}x}\ln x + \frac{\mathrm{y}}{x}\right) + \mathrm{y}^x\left(\ln \mathrm{y} + \frac{x}{\mathrm{y}}\frac{\mathrm{dy}}{\mathrm{d}x}\right) = 0\)
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