Find the derivative of y w.r.t. x in each of the following :
`sqrtx+sqrty=20`

Answer

To find the derivative of \( y \) with respect to \( x \) for the equation \( \sqrt{x} + \sqrt{y} = 20 \), we will use implicit differentiation. Here’s how to solve it step by step:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ \sqrt{x} + \sqrt{y} = 20 \]
Now, we differentiate both sides with respect to \( x \).

### Step 2: Apply the differentiation rules
Using the chain rule and the power rule, we differentiate each term:
- The derivative of \( \sqrt{x} \) is:
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]
- The derivative of \( \sqrt{y} \) with respect to \( x \) is:
\[ \frac{d}{dx}(\sqrt{y}) = \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} \]
- The derivative of the constant \( 20 \) is \( 0 \).

Putting it all together, we have:
\[ \frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = 0 \]

### 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, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} \]

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