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
Here are the derivatives using the product rule for each function: Part (i): Given f(x) = (3x^2 + 2)(5x - 3) Step 1: Identify u and v. Let u = 3x^2 + 2 and v = 5x - 3. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(3x^2 + 2) = 6x v' = (d)/(dx)(5x - 3) = 5 Step 3: Apply the product rule f'(x) = u'v + uv'. f'(x) = (6x)(5x - 3) + (3x^2 + 2)(5) Step 4: Expand and simplify the expression. f'(x) = 30x^2 - 18x + 15x^2 + 10 f'(x) = 45x^2 - 18x + 10 The derivative is 45x^2 - 18x + 10. Part (ii): Given y = e^2x x Step 1: Identify u and v. Let u = e^2x and v = x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(e^2x) = 2e^2x v' = (d)/(dx)( x) = x Step 3: Apply the product rule (dy)/(dx) = u'v + uv'. (dy)/(dx) = (2e^2x)( x) + (e^2x)( x) Step 4: Factor out common terms. (dy)/(dx) = e^2x(2 x + x) The derivative is e^2x(2 x + x). Part (iii): Given y = e^2x x (assuming x refers to the natural logarithm, x) Step 1: Identify u and v. Let u = e^2x and v = x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(e^2x) = 2e^2x v' = (d)/(dx)( x) = (1)/(x) Step 3: Apply the product rule (dy)/(dx) = u'v + uv'. (dy)/(dx) = (2e^2x)( x) + (e^2x)((1)/(x)) Step 4: Factor out common terms. (dy)/(dx) = e^2x(2 x + (1)/(x)) The derivative is e^2x(2 x + (1)/(x)). Send me the next one 📸