Find `dy/dx if siny-x=7`

Answer

To find \(\frac{dy}{dx}\) from the equation \(\sin y - x = 7\), we will follow these steps:

### Step 1: Start with the given equation
We have the equation:
\[ \sin y - x = 7 \]

### Step 2: Rewrite the equation
We can rewrite the equation as:
\[ \sin y = x + 7 \]

### Step 3: Differentiate both sides with respect to \(x\)
Now, we differentiate both sides of the equation with respect to \(x\):
\[ \frac{d}{dx}(\sin y) = \frac{d}{dx}(x + 7) \]

### Step 4: Apply the chain rule
Using the chain rule on the left side:
\[ \cos y \cdot \frac{dy}{dx} = 1 \]
The right side differentiates to:
\[ \frac{d}{dx}(x) + \frac{d}{dx}(7) = 1 + 0 = 1 \]

### Step 5: Solve for \(\frac{dy}{dx}\)
Now we can solve for \(\frac{dy}{dx}\):
\[ \cos y \cdot \frac{dy}{dx} = 1 \]
Dividing both sides by \(\cos y\):
\[ \frac{dy}{dx} = \frac{1}{\cos y} \]

### Step 6: Express in terms of secant
We know that \(\frac{1}{\cos y} = \sec y\), so we can write:
\[ \frac{dy}{dx} = \sec y \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \sec y \]

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