If 3^x y^x = x^{3y},then the value of frac{dy | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the equation: $3^x y^x = x^{3y}$.Taking the natural logarithm on both sides:$\ln(3^x y^x) = \ln(x^{3y})$$x \ln 3 + x \ln y = 3y \ln x$.Diffe

Vedclass · Accepted Answer

$\frac{1}{3}$

Vedclass · Answer

(D) Given the equation: $3^x y^x = x^{3y}$.Taking the natural logarithm on both sides:$\ln(3^x y^x) = \ln(x^{3y})$$x \ln 3 + x \ln y = 3y \ln x$.Differentiating both sides with respect to $x$:$\frac{d}{dx}(x \ln 3) + \frac{d}{dx}(x \ln y) = \frac{d}{dx}(3y \ln x)$$\ln 3 + (\ln y + x \cdot \frac{1}{y} \cdot \frac{dy}{dx}) = 3 \cdot \frac{dy}{dx} \cdot \ln x + 3y \cdot \frac{1}{x}$.At $x = 1$,we first find $y$ from the original equation:$3^1 y^1 = 1^{3y} \implies 3y = 1 \implies y = \frac{1}{3}$.Substituting $x = 1$ and $y = \frac{1}{3}$ into the differentiated equation:$\ln 3 + \ln(\frac{1}{3}) + 1 \cdot \frac{1}{1/3} \cdot \frac{dy}{dx} = 3 \cdot \frac{dy}{dx} \cdot \ln 1 + 3(\frac{1}{3}) \cdot \frac{1}{1}$.Since $\ln(\frac{1}{3}) = -\ln 3$ and $\ln 1 = 0$:$\ln 3 - \ln 3 + 3 \frac{dy}{dx} = 0 + 1$.$3 \frac{dy}{dx} = 1$.$\frac{dy}{dx} = \frac{1}{3}$.

If $3^x y^x = x^{3y}$,then the value of $\frac{dy}{dx}$ at $x = 1$ is:

Similar Questions

If $x^2+y^2=t-\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$,then $\frac{dy}{dx}=$

If $\sin x \sqrt{\cos y} - \cos y \sqrt{\sin x} = 0$,then $\frac{dy}{dx} = $

Let $f(x), x \in [0, \infty)$ be a non-negative continuous function. If $f'(x) \cos x \le f(x) \sin x$ for all $x \ge 0$,then the value of $f(2\pi)$ is equal to

If $\log y = y^{\log x}$,then $\frac{dy}{dx} = $

If $\ln (x + y) = 2xy$,then $y'(0) =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module