Find `dy/dx if x-ay=bx^2`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x - ay = bx^2\), we will use implicit differentiation. Here’s the step-by-step solution:

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

### Step 2: Apply the differentiation rules
- The derivative of \(x\) with respect to \(x\) is \(1\).
- The term \(-ay\) involves \(y\), which is a function of \(x\). We apply the product rule here:
\[ \frac{d}{dx}(-ay) = -a \frac{dy}{dx} \]
- The derivative of \(bx^2\) is:
\[ \frac{d}{dx}(bx^2) = b \cdot 2x = 2bx \]

So, differentiating both sides gives us:
\[ 1 - a \frac{dy}{dx} = 2bx \]

### Step 3: Rearrange the equation to solve for \(\frac{dy}{dx}\)
Now, we rearrange the equation to isolate \(\frac{dy}{dx}\):
\[ -a \frac{dy}{dx} = 2bx - 1 \]
Dividing both sides by \(-a\):
\[ \frac{dy}{dx} = \frac{2bx - 1}{-a} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{1 - 2bx}{a} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{1 - 2bx}{a} \]

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