Find `dy/dx if 2x-10y=logx`

Answer

To find \(\frac{dy}{dx}\) for the equation \(2x - 10y = \log x\), we will differentiate both sides of the equation with respect to \(x\).

### Step-by-step Solution:

1. **Start with the given equation:**
\[ 2x - 10y = \log x \]

2. **Differentiate both sides with respect to \(x\):**
- The derivative of \(2x\) is \(2\).
- The derivative of \(-10y\) is \(-10 \frac{dy}{dx}\) (using the chain rule).
- The derivative of \(\log x\) is \(\frac{1}{x}\).

So, differentiating both sides gives us:
\[ 2 - 10 \frac{dy}{dx} = \frac{1}{x} \]

3. **Rearrange the equation to isolate \(\frac{dy}{dx}\):**
\[ -10 \frac{dy}{dx} = \frac{1}{x} - 2 \]

4. **Multiply both sides by \(-1\):**
\[ 10 \frac{dy}{dx} = -\left(\frac{1}{x} - 2\right) \]

5. **Simplify the right side:**
\[ 10 \frac{dy}{dx} = 2 - \frac{1}{x} \]

6. **Divide both sides by \(10\) to solve for \(\frac{dy}{dx}\):**
\[ \frac{dy}{dx} = \frac{2 - \frac{1}{x}}{10} \]

7. **Combine the terms in the numerator:**
\[ \frac{dy}{dx} = \frac{2x - 1}{10x} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{2x - 1}{10x} \]