TS EAMCET · Maths · Differentiation
If \(3^x y^x=x^{3 y}\) then the value of \(\frac{d y}{d x}\) at \(x=1\) is
- A \(-3\)
- B \(3\)
- C \(-\frac{1}{3}\)
- D \(\frac{1}{3}\)
Answer & Solution
Correct Answer
(D) \(\frac{1}{3}\)
Step-by-step Solution
Detailed explanation
\(3^x y^x = x^{3y}\) \(x\ln(3y) = 3y\ln x\) At \(x=1\): \(3^1 y^1 = 1^{3y} \implies 3y = 1 \implies y = \frac{1}{3}\) Differentiating implicitly: \(\ln(3y) + \frac{x}{y}\frac{dy}{dx} = 3\ln x\frac{dy}{dx} + \frac{3y}{x}\) Substitute \(x=1, y=\frac{1}{3}\):…
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