If `x^(y)=e^(x-3)` then dy/dx is equal to which one of the following ?

Options

A. `((x-y))/((xlogx))`
B. `(y)/((1+logx))`
C. `((x+y))/((1+logx))`
D. `((logx))/((1+logx)^(2))`

Answer

To find \( \frac{dy}{dx} \) for the equation \( x^y = e^{x-3} \), we will follow these steps:

### Step 1: Take the natural logarithm of both sides
We start by taking the natural logarithm of both sides of the equation:
\[ \ln(x^y) = \ln(e^{x-3}) \]

### Step 2: Simplify using logarithmic properties
Using the properties of logarithms, we can simplify both sides:
\[ y \ln(x) = x - 3 \]

### Step 3: Differentiate both sides with respect to \( x \)
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y \ln(x)) = \frac{d}{dx}(x - 3) \]
Using the product rule on the left side, we get:
\[ \frac{dy}{dx} \ln(x) + y \frac{1}{x} = 1 \]

### Step 4: Rearrange the equation to isolate \( \frac{dy}{dx} \)
Now, we can rearrange the equation to solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} \ln(x) = 1 - \frac{y}{x} \]
\[ \frac{dy}{dx} = \frac{1 - \frac{y}{x}}{\ln(x)} \]

### Step 5: Simplify the expression
We can rewrite the expression to make it clearer:
\[ \frac{dy}{dx} = \frac{x - y}{x \ln(x)} \]

Thus, the final answer is:
\[ \frac{dy}{dx} = \frac{x - y}{x \ln(x)} \]