if `sqrtx+sqrty=4`, then `(dy)/(dx)` is :

Options

A. `-sqrt(x/y)`
B. `-sqrt(y/x)`
C. `sqrt(x/y)`
D. `sqrt(y/x)`

Answer

To find \(\frac{dy}{dx}\) given the equation \(\sqrt{x} + \sqrt{y} = 4\), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ \sqrt{x} + \sqrt{y} = 4 \]
Now, we differentiate both sides with respect to \(x\).

### Step 2: Apply the differentiation
Using the chain rule, we differentiate:
- The derivative of \(\sqrt{x}\) is \(\frac{1}{2\sqrt{x}}\).
- The derivative of \(\sqrt{y}\) is \(\frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx}\) (using the chain rule since \(y\) is a function of \(x\)).

So, differentiating gives us:
\[ \frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = 0 \]

### Step 3: Isolate \(\frac{dy}{dx}\)
Now, we need to isolate \(\frac{dy}{dx}\):
\[ \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = -\frac{1}{2\sqrt{x}} \]

Multiplying both sides by \(2\sqrt{y}\) gives:
\[ \frac{dy}{dx} = -\frac{2\sqrt{y}}{2\sqrt{x}} = -\frac{\sqrt{y}}{\sqrt{x}} \]

### Final Result
Thus, we have:
\[ \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} \]