This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — let's solve questions 3, 4, and 5. Question 3 a) Differentiate with respect to x: ((1+x^2))/(x^2) Step 1: Identify the function as a quotient and define u and v. Let f(x) = (u)/(v), where u = (1+x^2) and v = x^2. Step 2: Find the derivatives of u and v. Using the chain rule for u: (du)/(dx) = (1)/(1+x^2) · (d)/(dx)(1+x^2) = (2x)/(1+x^2) Using the power rule for v: (dv)/(dx) = 2x Step 3: Apply the quotient rule, (d)/(dx)((u)/(v)) = (vdu)/(dx) - u(dv)/(dx)v^2. f'(x) = (x^2(2x)/(1+x^2)) - (1+x^2)(2x)(x^2)^2 f'(x) = (2x^3)/(1+x^2) - 2x(1+x^2)x^4 Step 4: Simplify the expression. Factor out 2x from the numerator: f'(x) = (2x(x^2)/(1+x^2) - (1+x^2)\