If `sqrtx+sqrty=5,"find "dy/dx` at (4, 9).

Answer

To find \(\frac{dy}{dx}\) at the point (4, 9) for the equation \(\sqrt{x} + \sqrt{y} = 5\), we will use implicit differentiation. Here is the step-by-step solution:

### Step 1: Differentiate both sides of the equation
Given the equation:
\[ \sqrt{x} + \sqrt{y} = 5 \]
we differentiate both sides with respect to \(x\).

### Step 2: Apply the differentiation
Using the chain rule, we differentiate:
\[ \frac{d}{dx}(\sqrt{x}) + \frac{d}{dx}(\sqrt{y}) = \frac{d}{dx}(5) \]
This gives us:
\[ \frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = 0 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to solve for \(\frac{dy}{dx}\):
\[ \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = -\frac{1}{2\sqrt{x}} \]
Multiplying both sides by \(2\sqrt{y}\) to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{2\sqrt{y}}{2\sqrt{x}} = -\frac{\sqrt{y}}{\sqrt{x}} \]

### Step 4: Substitute the point (4, 9)
Now we substitute \(x = 4\) and \(y = 9\) into the equation:
\[ \frac{dy}{dx} = -\frac{\sqrt{9}}{\sqrt{4}} = -\frac{3}{2} \]

### Final Result
Thus, the value of \(\frac{dy}{dx}\) at the point (4, 9) is:
\[ \frac{dy}{dx} = -\frac{3}{2} \]

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