Find `dy/dx if x=y sinx`

Answer

To find \(\frac{dy}{dx}\) given the equation \(x = y \sin x\), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides with respect to \(x\)
We start with the equation:
\[ x = y \sin x \]
Differentiating both sides with respect to \(x\), we have:
\[ \frac{d}{dx}(x) = \frac{d}{dx}(y \sin x) \]

### Step 2: Apply the differentiation
The left-hand side differentiates to:
\[ 1 \]
For the right-hand side, we need to apply the product rule. The product rule states that if you have a product of two functions \(u\) and \(v\), then:
\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]
Here, let \(u = y\) and \(v = \sin x\). Thus:
\[ \frac{d}{dx}(y \sin x) = y \frac{d}{dx}(\sin x) + \sin x \frac{dy}{dx} \]
This gives us:
\[ \frac{d}{dx}(y \sin x) = y \cos x + \sin x \frac{dy}{dx} \]

### Step 3: Set the derivatives equal
Now we can set the derivatives equal:
\[ 1 = y \cos x + \sin x \frac{dy}{dx} \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Rearranging the equation to isolate \(\frac{dy}{dx}\):
\[ \sin x \frac{dy}{dx} = 1 - y \cos x \]
Now, divide both sides by \(\sin x\):
\[ \frac{dy}{dx} = \frac{1 - y \cos x}{\sin x} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{1 - y \cos x}{\sin x} \]

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