Find slope of tangent to the curve if equation is `x^2 + y^2 = 9`

Answer

To find the slope of the tangent to the curve given by the equation \(x^2 + y^2 = 9\), we can follow these steps:

### Step 1: Differentiate the equation
We start with the equation of the curve:
\[ x^2 + y^2 = 9 \]
We will differentiate both sides of the equation with respect to \(x\).

### Step 2: Apply differentiation
Differentiating \(x^2\) gives:
\[ \frac{d}{dx}(x^2) = 2x \]
Differentiating \(y^2\) using implicit differentiation gives:
\[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]
So, differentiating the entire equation, we have:
\[ 2x + 2y \frac{dy}{dx} = 0 \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Now, we can isolate \(\frac{dy}{dx}\):
\[ 2y \frac{dy}{dx} = -2x \]
Dividing both sides by \(2y\):
\[ \frac{dy}{dx} = -\frac{x}{y} \]

### Step 4: Interpret the result
The slope of the tangent line to the curve at any point \((a, b)\) on the curve is given by:
\[ \text{slope} = \frac{dy}{dx} = -\frac{a}{b} \]
where \((a, b)\) is a point on the curve.

### Step 5: Conclusion
Thus, the slope of the tangent to the curve \(x^2 + y^2 = 9\) at the point \((a, b)\) is:
\[ -\frac{a}{b} \]