Find the slope of tangent to the curve if equation of the curve is `y^2=4x` at ( 5, 6)

Answer

To find the slope of the tangent to the curve given by the equation \( y^2 = 4x \) at the point \( (5, 6) \), we will follow these steps:

### Step 1: Differentiate the equation of the curve
The equation of the curve is:
\[ y^2 = 4x \]
To find the slope of the tangent, we need to differentiate both sides with respect to \( x \).

### Step 2: Apply implicit differentiation
Differentiating both sides, we get:
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4x) \]
Using the chain rule on the left side, we have:
\[ 2y \frac{dy}{dx} = 4 \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y} \]

### Step 4: Substitute the given point into the derivative
We need to find the slope at the point \( (5, 6) \). Here, \( y = 6 \):
\[ \frac{dy}{dx} = \frac{2}{6} = \frac{1}{3} \]

### Conclusion
Thus, the slope of the tangent to the curve \( y^2 = 4x \) at the point \( (5, 6) \) is:
\[ \frac{1}{3} \]

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