If `x^2+y^2=25 " then find " dy/dx.`

Options

A. `dy/dx =y`
B. `dy/dx =x`
C. `dy/dx =-y/x`
D. `dy/dx =-x/y`

Explanation

To find \( \frac{dy}{dx} \) given the equation \( x^2 + y^2 = 25 \), we will use implicit differentiation. Here are the steps to solve the problem:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ x^2 + y^2 = 25 \]
Now, we differentiate both sides with respect to \( x \).

### Step 2: Apply differentiation
Differentiating \( x^2 \) gives:
\[ \frac{d}{dx}(x^2) = 2x \]
For \( y^2 \), we apply the chain rule:
\[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]
The right side, being a constant (25), differentiates to:
\[ \frac{d}{dx}(25) = 0 \]

### Step 3: Combine the results
Putting it all together, we have:
\[ 2x + 2y \frac{dy}{dx} = 0 \]

### Step 4: Isolate \( \frac{dy}{dx} \)
Now, we need to isolate \( \frac{dy}{dx} \):
\[ 2y \frac{dy}{dx} = -2x \]
Dividing both sides by \( 2y \):
\[ \frac{dy}{dx} = -\frac{2x}{2y} \]
This simplifies to:
\[ \frac{dy}{dx} = -\frac{x}{y} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{x}{y} \]

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