Answer:
To find \(\frac{dy}{dx}\) in terms of \(y\) for the equation \(3x - \tan(y) = 4\), you can use implicit differentiation. Implicit differentiation allows you to find the derivative of \(y\) with respect to \(x\) when you have an equation involving both \(x\) and \(y\).
Start by differentiating both sides of the equation with respect to \(x\):
\(\frac{d}{dx}(3x - \tan(y)) = \frac{d}{dx}(4)\)
Now, differentiate each term on the left side using the chain rule for \(\tan(y)\):
\(3 \cdot \frac{d}{dx}(x) - \frac{d}{dx}(\tan(y)) = 0\)
Since \(\frac{d}{dx}(x)\) is just 1, we have:
\(3 - \frac{d}{dx}(\tan(y)) = 0\)
Now, isolate \(\frac{d}{dx}(\tan(y))\) by moving the 3 to the other side:
\(\frac{d}{dx}(\tan(y)) = 3\)
So, \(\frac{dy}{dx}\) in terms of \(y\) is:
\(\frac{dy}{dx} = 3\)
Explanation: