Find `dy/dx if siny=3x`

Answer

To find \(\frac{dy}{dx}\) given the equation \(\sin y = 3x\), we will use implicit differentiation. Here are the steps:

### Step 1: Differentiate both sides of the equation
Start with the equation:
\[ \sin y = 3x \]
Now, differentiate both sides with respect to \(x\).

### Step 2: Apply the chain rule
Using the chain rule for the left side:
\[ \frac{d}{dx}(\sin y) = \cos y \cdot \frac{dy}{dx} \]
And for the right side:
\[ \frac{d}{dx}(3x) = 3 \]

### Step 3: Set the derivatives equal
Now, we can set the derivatives equal to each other:
\[ \cos y \cdot \frac{dy}{dx} = 3 \]

### Step 4: Solve for \(\frac{dy}{dx}\)
To isolate \(\frac{dy}{dx}\), divide both sides by \(\cos y\):
\[ \frac{dy}{dx} = \frac{3}{\cos y} \]

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

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