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PUNE MCA Exam 2015 Question Paper with answers for online practice

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Question : 19 of 59

Marks: +1, -0

If

x^y=e^{x-y}

\frac{dy}{dx}

at (1, 1) ?

1
2
0
4

Solution:

Concept:

Implicit differentiation using logarithms.

Explanation:

Take natural log of both sides:

y \ln x = x - y

.

Differentiate implicitly:

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

.

Bring

\frac{dy}{dx}

terms together:

\frac{dy}{dx} (\ln x + 1) = 1 - \frac{y}{x}

.

Thus

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

.

Evaluate at

(1,1)

:

\ln 1 = 0

, so

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

.

Answer:

\frac{dy}{dx}

at

(1,1)

is

0

.

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