Share

Books Shortlist

# If X Y = E X − Y , Then D Y D X is - CBSE (Arts) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

If $x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}$ is __________ .

• $\frac{1 + x}{1 + \log x}$

• $\frac{1 - \log x}{1 + \log x}$

• not defined

• $\frac{\log x}{\left( 1 + \log x \right)^2}$

#### Solution

$\frac{\log x}{\left( 1 + \log x \right)^2}$

$\text{ We have,} x^y = e^{x - y}$
$\text{ Taking log on both sides we get },$
$\Rightarrow y \log x = \left( x - y \right) \log_e e$
$\Rightarrow y \log x = x - y$
$\Rightarrow y \log x + y = x$
$\Rightarrow y\left( 1 + \log x \right) = x$
$\Rightarrow y = \frac{x}{1 + \log x}$

$\Rightarrow \frac{dy}{dx} = \frac{\left( 1 + \log x \right) \times 1 - x \times \left( 0 + \frac{1}{x} \right)}{\left( 1 + \log x \right)^2}$
$\Rightarrow \frac{dy}{dx} = \frac{1 + \log x - 1}{\left( 1 + \log x \right)^2}$
$\Rightarrow \frac{dy}{dx} = \frac{\log x}{\left( 1 + \log x \right)^2}$

Is there an error in this question or solution?

#### Video TutorialsVIEW ALL [1]

Solution If X Y = E X − Y , Then D Y D X is Concept: Simple Problems on Applications of Derivatives.
S