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

Answer

To find \(\frac{dy}{dx}\) for the equation \(4x^2 - y = \sin x\), we will follow these steps:

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

### Step 2: Apply the differentiation rules
Using the rules of differentiation:
- The derivative of \(4x^2\) is \(8x\) (using the power rule).
- The derivative of \(y\) with respect to \(x\) is \(\frac{dy}{dx}\).
- The derivative of \(\sin x\) is \(\cos x\).

So, differentiating both sides gives us:
\[ \frac{d}{dx}(4x^2) - \frac{d}{dx}(y) = \frac{d}{dx}(\sin x) \]
This results in:
\[ 8x - \frac{dy}{dx} = \cos x \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Now, we need to isolate \(\frac{dy}{dx}\):
\[ -\frac{dy}{dx} = \cos x - 8x \]
Multiplying both sides by -1 gives:
\[ \frac{dy}{dx} = 8x - \cos x \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 8x - \cos x \]

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