Find dydx if, xy=ex - y - Mathematics and Statistics

Question

Find $\frac{dy}{dx}$ if, $x^{y} = e^{x - y}$

Sum

Solution

$x^{y} = e^{x - y}$

Taking logarithm of both sides, we get

y log x = (x - y) log e = x - y

∴ y log x + y = x

∴ y(1 + log x) = x

∴ y = $\frac{x}{1 + \log x}$

Differentiating both sides w.r.t. x, we get

$\frac{dy}{dx} = \frac{d}{dx} [\frac{x}{1 + \log x}]$

∴ $\frac{dy}{dx} = \frac{(1 + \log x) \frac{d}{dx} (x) - x \frac{d}{dx} (1 + \log x)}{{(1 + \log x)}^{2}}$

$= \frac{(1 + \log x) \times 1 - x \times (\frac{1}{x})}{{(1 + \log x)}^{2}}$

$= \frac{1 + \log x - 1}{{(1 + \log x)}^{2}}$

∴ $\frac{dy}{dx} = \frac{\log x}{{(1 + \log x)}^{2}}$

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Derivatives of Implicit Functions

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Q 2. 2)Q 2. 1)Q 2. 3)

Chapter 3: Differentiation - EXERCISE 3.4 [Page 95]

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Balbharati Mathematics and Statistics 1 (Commerce) [English] 12 Standard HSC Maharashtra State Board

Chapter 3 Differentiation
EXERCISE 3.4 | Q 2. 2) | Page 95

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Find dydx if, xy=ex - y - Mathematics and Statistics

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