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
Benoni, let's knock this out. To find the derivative of the given functions, we will use the chain rule and the product rule. Chain Rule: If y = [f(x)]^n, then (dy)/(dx) = n[f(x)]^n-1 · f'(x). Product Rule: If y = u(x)v(x), then (dy)/(dx) = u'(x)v(x) + u(x)v'(x). i) y = (2x^3 - 3x + 5)^4 Step 1: Apply the chain rule. Let f(x) = 2x^3 - 3x + 5 and n=4. (dy)/(dx) = 4(2x^3 - 3x + 5)^4-1 · (d)/(dx)(2x^3 - 3x + 5) Step 2: Differentiate the inner function. (d)/(dx)(2x^3 - 3x + 5) = 6x^2 - 3 Step 3: Substitute the derivative of the inner function back into the chain rule formula. (dy)/(dx) = 4(2x^3 - 3x + 5)^3 (6x^2 - 3) (dy)/(dx) = 12(2x^2 - 1)(2x^3 - 3x + 5)^3 The derivative is 12(2x^2 - 1)(2x^3 - 3x + 5)^3. ii) y = (3x^3 + 7)^3 Step 1: Apply the chain rule. Let f(x) = 3x^3 + 7 and n=3. (dy)/(dx) = 3(3x^3 + 7)^3-1 · (d)/(dx)(3x^3 + 7) Step 2: Differentiate the inner function. (d)/(dx)(3x^3 + 7) = 9x^2 Step 3: Substitute the derivative of the inner function back into the chain rule formula. (dy)/(dx) = 3(3x^3 + 7)^2 (9x^2) (dy)/(dx) = 27x^2(3x^3 + 7)^2 The derivative is 27x^2(3x^3 + 7)^2. iii) y = (x^4 - 4)^4 Step 1: Apply the chain rule. Let f(x) = x^4 - 4 and n=4. (dy)/(dx) = 4(x^4 - 4)^4-1 · (d)/(dx)(x^4 - 4) Step 2: Differentiate the inner function. (d)/(dx)(x^4 - 4) = 4x^3 Step 3: Substitute the derivative of the inner function back into the chain rule formula. (dy)/(dx) = 4(x^4 - 4)^3 (4x^3) (dy)/(dx) = 16x^3(x^4 - 4)^3 The derivative is 16x^3(x^4 - 4)^3. iv) y = (2x-1)^2 (3x^2+1)^2 Step 1: Apply the product rule. Let u = (2x-1)^2 and v = (3x^2+1)^2. (dy)/(dx) = u'v + uv' Step 2: Find u' using the chain rule. u' = (d)/(dx)(2x-1)^2 = 2(2x-1)^2-1 · (d)/(dx)(2x-1) = 2(2x-1)(2) = 4(2x-1) Step 3: Find v' using the chain rule. v' = (d)/(dx)(3x^2+1)^2 = 2(3x^2+1)^2-1 · (d)/(dx)(3x^2+1) = 2(3x^2+1)(6x) = 12x(3x^2+1) Step 4: Substitute u, v, u', v' into the product rule formula. (dy)/(dx) = 4(2x-1)^2 + (2x-1)^2[12x(3x^2+1)] Step 5: Factor out common terms, which are 4(2x-1)(3x^2+1). (dy)/(dx) = 4(2x-1)(3x^2+1) [(3x^2+1) + 3x(2x-1)] Step 6: Simplify the expression inside the brackets. (3x^2+1) + 3x(2x-1) = 3x^2+1 + 6x^2-3x = 9x^2-3x+1 Step 7: Write the final simplified derivative. (dy)/(dx) = 4(2x-1)(3x^2+1)(9x^2-3x+1) The derivative is 4(2x-1)(3x^2+1)(9x^2-3x+1). Send me the next one 📸