This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
Let's find the derivative (dy)/(dx) for the given function. The function is y = (2x) + ^2(3x) + ^2(x). We will differentiate each term separately using the chain rule. Step 1: Differentiate the first term, (2x). Let u = 2x. Then (du)/(dx) = 2. The derivative of (u) is ^2(u) · (du)/(dx). (d)/(dx)((2x)) = ^2(2x) · 2 = 2^2(2x) Step 2: Differentiate the second term, ^2(3x). This can be written as ((3x))^2. Let u = (3x). Then (d)/(dx)(u^2) = 2u (du)/(dx). Now, find (du)/(dx) for u = (3x). Let v = 3x. Then (dv)/(dx) = 3. The derivative of (v) is (v) · (dv)/(dx). So, (du)/(dx) = (3x) · 3 = 3(3x). Now substitute back into the derivative of u^2: (d)/(dx)(^2(3x)) = 2(3x) · (3(3x)) = 6(3x)(3x) Using the identity (2) = 2, we can simplify this: 6(3x)(3x) = 3 · (2(3x)(3x)) = 3(2 · 3x) = 3(6x) Step 3: Differentiate the third term, ^2(x). This can be written as ((x))^2. Let w = (x). Then (d)/(dx)(w^2) = 2w (dw)/(dx). Now, find (dw)/(dx) for w = (x). (dw)/(dx) = -(x) Now substitute back into the derivative of w^2: (d)/(dx)(^2(x)) = 2(x) · (-(x)) = -2(x)(x) Using the identity (2) = 2, we can simplify this: -2(x)(x) = -(2x) Step 4: Combine the derivatives of all terms. (dy)/(dx) = (d)/(dx)((2x)) + (d)/(dx)(^2(3x)) + (d)/(dx)(^2(x)) (dy)/(dx) = 2^2(2x) + 3(6x) - (2x) The final answer is (dy)/(dx) = 2^2(2x) + 3(6x) - (2x). 3 done, 2 left today. You're making progress.