Find `dy/dx if 3x^2-4y= cosx`

Answer

To find \(\frac{dy}{dx}\) for the equation \(3x^2 - 4y = \cos x\), we will use implicit differentiation. Here are the step-by-step calculations:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ 3x^2 - 4y = \cos x \]
Now, we differentiate both sides with respect to \(x\).

### Step 2: Apply the differentiation rules
Using the rules of differentiation:
- The derivative of \(3x^2\) is \(6x\) (using the power rule).
- The derivative of \(-4y\) is \(-4 \frac{dy}{dx}\) (since \(y\) is a function of \(x\)).
- The derivative of \(\cos x\) is \(-\sin x\).

So, differentiating both sides gives us:
\[ \frac{d}{dx}(3x^2) - \frac{d}{dx}(4y) = \frac{d}{dx}(\cos x) \]
This results in:
\[ 6x - 4\frac{dy}{dx} = -\sin x \]

### Step 3: Rearrange the equation to isolate \(\frac{dy}{dx}\)
Now, we want to isolate \(\frac{dy}{dx}\). We can rearrange the equation:
\[ 6x + \sin x = 4\frac{dy}{dx} \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Now, divide both sides by 4 to solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{6x + \sin x}{4} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{6x + \sin x}{4} \]

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