Find `dy/dx at (6,1) if y^2=4ax`

Answer

To find \(\frac{dy}{dx}\) at the point \((6, 1)\) for the equation \(y^2 = 4ax\), we will follow these steps:

### Step 1: Differentiate the equation
We start with the equation:
\[ y^2 = 4ax \]
To differentiate both sides with respect to \(x\), we apply implicit differentiation. The left side becomes:
\[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]
And the right side becomes:
\[ \frac{d}{dx}(4ax) = 4a \cdot \frac{d}{dx}(x) = 4a \]
So, we have:
\[ 2y \frac{dy}{dx} = 4a \]

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

### Step 3: Substitute the point \((6, 1)\)
Next, we need to find the value of \(a\) using the point \((6, 1)\). We substitute \(x = 6\) and \(y = 1\) into the original equation:
\[ 1^2 = 4a \cdot 6 \]
This simplifies to:
\[ 1 = 24a \]
From this, we can solve for \(a\):
\[ a = \frac{1}{24} \]

### Step 4: Substitute \(a\) and \(y\) into \(\frac{dy}{dx}\)
Now we substitute \(a\) and \(y\) into the expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{2 \cdot \frac{1}{24}}{1} = \frac{2}{24} = \frac{1}{12} \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) at the point \((6, 1)\) is:
\[ \frac{dy}{dx} = \frac{1}{12} \]

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