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
Okay, Nurse, let's find the derivative of this function. The given function is y = 5x^2 e^4x+12. We need to find (dy)/(dx). This requires the product rule and the chain rule. Step 1: Identify the two functions in the product. Let u = 5x^2 and v = e^4x+12. The product rule states that if y = uv, then (dy)/(dx) = u'v + uv'. Step 2: Differentiate u with respect to x. u = 5x^2 u' = (d)/(dx)(5x^2) = 5 · 2x = 10x Step 3: Differentiate v with respect to x using the chain rule. v = e^4x+12 The derivative of e^f(x) is e^f(x) · f'(x). Here, f(x) = 4x+12. f'(x) = (d)/(dx)(4x+12) = 4 So, v' = (d)/(dx)(e^4x+12) = e^4x+12 · 4 = 4e^4x+12 Step 4: Apply the product rule formula. Substitute u, u', v, and v' into the product rule formula: (dy)/(dx) = u'v + uv'. (dy)/(dx) = (10x)(e^4x+12) + (5x^2)(4e^4x+12) Step 5: Simplify the expression. (dy)/(dx) = 10x e^4x+12 + 20x^2 e^4x+12 Factor out the common terms, which are 10x e^4x+12. (dy)/(dx) = 10x e^4x+12 (1 + 2x) The final answer is 10xe^4x+12(1 + 2x). 3 done, 2 left today. You're making progress.