Find `dy/dx if sin(x-y)=3x`

Answer

To find \( \frac{dy}{dx} \) given the equation \( \sin(x - y) = 3x \), we will differentiate both sides with respect to \( x \). Here’s a step-by-step solution:

### Step 1: Differentiate both sides
We start with the equation:
\[ \sin(x - y) = 3x \]
Now we differentiate both sides with respect to \( x \).

### Step 2: Apply the chain rule
Using the chain rule on the left side, we have:
\[ \frac{d}{dx}[\sin(x - y)] = \cos(x - y) \cdot \frac{d}{dx}(x - y) \]
Now, differentiate \( x - y \):
\[ \frac{d}{dx}(x - y) = 1 - \frac{dy}{dx} \]
So, the left side becomes:
\[ \cos(x - y)(1 - \frac{dy}{dx}) \]

### Step 3: Differentiate the right side
Now differentiate the right side:
\[ \frac{d}{dx}[3x] = 3 \]

### Step 4: Set the derivatives equal
Now we equate the derivatives from both sides:
\[ \cos(x - y)(1 - \frac{dy}{dx}) = 3 \]

### Step 5: Expand and rearrange
Expanding the left side gives:
\[ \cos(x - y) - \cos(x - y) \frac{dy}{dx} = 3 \]
Now, rearranging this equation to isolate \( \frac{dy}{dx} \):
\[ -\cos(x - y) \frac{dy}{dx} = 3 - \cos(x - y) \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Now, divide both sides by \( -\cos(x - y) \):
\[ \frac{dy}{dx} = \frac{3 - \cos(x - y)}{\cos(x - y)} \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{3 - \cos(x - y)}{\cos(x - y)} \]

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