If y=c`e^(x//(x-a))`, then `(dy)/(dx)` equals

Options

A. `a(x-a)^(2)`
B. `-(ay)/((x-a)^(2))`
C. `a^(2)(x-a)^(2)`
D. `a(x-a)`

Answer

To find the derivative of the function \( y = c e^{\frac{x}{x-a}} \), we will apply the chain rule and the quotient rule. Here is the step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = c e^{\frac{x}{x-a}} \]
where \( c \) is a constant.

### Step 2: Differentiate using the chain rule
Using the chain rule, the derivative of \( e^{u} \) (where \( u = \frac{x}{x-a} \)) is:
\[ \frac{dy}{dx} = c e^{u} \cdot \frac{du}{dx} \]
So we need to find \( \frac{du}{dx} \).

### Step 3: Find \( u \) and differentiate it using the quotient rule
Let \( u = \frac{x}{x-a} \). We will apply the quotient rule:
\[ \frac{du}{dx} = \frac{(x-a) \cdot \frac{d}{dx}(x) - x \cdot \frac{d}{dx}(x-a)}{(x-a)^2} \]
Calculating the derivatives:
- \( \frac{d}{dx}(x) = 1 \)
- \( \frac{d}{dx}(x-a) = 1 \)

Now substituting these into the quotient rule:
\[ \frac{du}{dx} = \frac{(x-a)(1) - x(1)}{(x-a)^2} = \frac{x - a - x}{(x-a)^2} = \frac{-a}{(x-a)^2} \]

### Step 4: Substitute back into the derivative
Now substituting \( \frac{du}{dx} \) back into the derivative of \( y \):
\[ \frac{dy}{dx} = c e^{\frac{x}{x-a}} \cdot \frac{-a}{(x-a)^2} \]

### Step 5: Simplify the expression
Thus, we have:
\[ \frac{dy}{dx} = -\frac{ac e^{\frac{x}{x-a}}}{(x-a)^2} \]

### Final Answer
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{ac e^{\frac{x}{x-a}}}{(x-a)^2} \]

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