Find `dy/dx if ax-by=sinx`

Answer

To find \(\frac{dy}{dx}\) for the equation \(ax - by = \sin x\), we will follow these steps:

### Step 1: Differentiate both sides with respect to \(x\)

We start with the equation:
\[ ax - by = \sin x \]

Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(ax) - \frac{d}{dx}(by) = \frac{d}{dx}(\sin x) \]

### Step 2: Apply the differentiation rules

Using the product rule and the chain rule, we differentiate each term:
- The derivative of \(ax\) is \(a\) (since \(a\) is a constant).
- The derivative of \(by\) is \(b \frac{dy}{dx}\) (using the chain rule).
- The derivative of \(\sin x\) is \(\cos x\).

Putting it all together, we have:
\[ a - b \frac{dy}{dx} = \cos x \]

### Step 3: Solve for \(\frac{dy}{dx}\)

Now, we need to isolate \(\frac{dy}{dx}\):
\[ -b \frac{dy}{dx} = \cos x - a \]

Dividing both sides by \(-b\):
\[ \frac{dy}{dx} = \frac{a - \cos x}{b} \]

### Final Answer

Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{a - \cos x}{b} \]

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