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UPSC NDA I 2022 Mathematics Solved Paper

Question : 69 of 120

Marks: +1, -0

x^{y}y^{x} = 1

\frac{dy}{dx}

(1, 1)

equal to?

Solution:

Concept:

Apply natural logarithms and implicit differentiation to find the derivative from

x^{y}y^{x}=1

Explanation:

Step 1: Take

\ln

on both sides.

\ln(x^{y}y^{x}) = \ln 1

Using

\ln(ab)=\ln a+\ln b

and

\ln a^n=n\ln a

y\ln x + x\ln y = 0

(since

\ln 1 =0

Step 2: Differentiate implicitly with respect to

x

\frac{d}{dx}(y\ln x)+\frac{d}{dx}(x\ln y)=0

Apply product rule:

y\cdot\frac{1}{x} + \ln x\cdot \frac{dy}{dx} + x\cdot\frac{1}{y}\cdot\frac{dy}{dx} + \ln y\cdot 1 =0

\frac{y}{x} + \ln y + \frac{dy}{dx}\left(\ln x + \frac{x}{y}\right) = 0

Step 3: Solve for

\frac{dy}{dx}

\frac{dy}{dx}\left(\frac{x}{y}+\ln x\right) = -\left(\frac{y}{x}+\ln y\right)

\frac{dy}{dx} = -\frac{\frac{y}{x}+\ln y}{\frac{x}{y}+\ln x}

Step 4: Substitute

(x,y)=(1,1)

x=1, y=1

\frac{1}{1}=1

\ln 1=0

Then

\frac{dy}{dx} = -\frac{1+0}{1+0} = -1

Answer:

The value of

\frac{dy}{dx}

(1,1)

-1

, which corresponds to option A.

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