Find `dy/dx if 2x-4y=sinx-x^2`

Answer

To find \(\frac{dy}{dx}\) for the equation \(2x - 4y = \sin(x) - x^2\), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ 2x - 4y = \sin(x) - x^2 \]
Now, we differentiate both sides with respect to \(x\).

### Step 2: Differentiate the left side
The left side is \(2x - 4y\). We differentiate each term:
- The derivative of \(2x\) is \(2\).
- The derivative of \(-4y\) is \(-4\frac{dy}{dx}\) (using the chain rule since \(y\) is a function of \(x\)).

So, the differentiation of the left side gives us:
\[ 2 - 4\frac{dy}{dx} \]

### Step 3: Differentiate the right side
The right side is \(\sin(x) - x^2\). We differentiate each term:
- The derivative of \(\sin(x)\) is \(\cos(x)\).
- The derivative of \(-x^2\) is \(-2x\).

So, the differentiation of the right side gives us:
\[ \cos(x) - 2x \]

### Step 4: Set the derivatives equal to each other
Now we set the derivatives from the left side equal to the derivatives from the right side:
\[ 2 - 4\frac{dy}{dx} = \cos(x) - 2x \]

### Step 5: Solve for \(\frac{dy}{dx}\)
Now, we need to isolate \(\frac{dy}{dx}\):
1. Rearrange the equation:
\[ -4\frac{dy}{dx} = \cos(x) - 2x - 2 \]
2. Divide both sides by \(-4\):
\[ \frac{dy}{dx} = \frac{-(\cos(x) - 2x - 2)}{4} \]
3. Simplify:
\[ \frac{dy}{dx} = \frac{2x + 2 - \cos(x)}{4} \]

### Final Result
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{2x + 2 - \cos(x)}{4} \]