Step 1 :Given the function \(y = 3\sin(x)\cos(x) + 2\tan(x)\)
Step 2 :We need to find the derivative of this function, \(\frac{d y}{d x}\)
Step 3 :Using the chain rule, the derivative of \(\sin(x)\cos(x)\) is \(-3\sin^2(x) + 3\cos^2(x)\) and the derivative of \(\tan(x)\) is \(2\tan^2(x) + 2\)
Step 4 :So, \(\frac{d y}{d x} = -3\sin^2(x) + 3\cos^2(x) + 2\tan^2(x) + 2\)
Step 5 :We then substitute \(x = \frac{\pi}{4}\) into the derivative
Step 6 :Which gives us \(\frac{d y}{d x} = 4\) at \(x = \frac{\pi}{4}\)
Step 7 :Final Answer: \(\boxed{4}\)