Find `dy/dx if sinx=x^3-y`

Answer

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

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

### Step 2: Apply differentiation rules
Using the differentiation rules:
- The derivative of \( \sin x \) is \( \cos x \).
- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} \).

So, differentiating both sides gives us:
\[ \frac{d}{dx}(\sin x) = \frac{d}{dx}(x^3 - y) \]
This results in:
\[ \cos x = 3x^2 - \frac{dy}{dx} \]

### Step 3: Rearrange the equation
Now, we need to isolate \( \frac{dy}{dx} \). We can do this by moving \( \frac{dy}{dx} \) to one side and the other terms to the opposite side:
\[ \frac{dy}{dx} = 3x^2 - \cos x \]

### Step 4: Final expression
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 3x^2 - \cos x \]