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
Hey Luke, good to see you again. Here are the derivatives of the given functions: 1. y = (2x^3 - 3x + 5)^4 Step 1: Apply the chain rule, (dy)/(dx) = n[f(x)]^n-1 · f'(x). Here, 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 expression. (dy)/(dx) = 4(2x^3 - 3x + 5)^3 (6x^2 - 3) (dy)/(dx) = 12(2x^3 - 3x + 5)^3 (2x^2 - 1) The derivative is 12(2x^3 - 3x + 5)^3 (2x^2 - 1). 2. y = (3x^3 + 7)^3 Step 1: Apply the chain rule. Here, 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. (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. 3. y = (x^4 - 4)^4 Step 1: Apply the chain rule. Here, 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. (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. 4. y = (2x - 1)^2 (3x^2 + 1)^2 Step 1: Apply the product rule, (dy)/(dx) = u'v + uv'. Let u = (2x - 1)^2 and v = (3x^2 + 1)^2. 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, u', v, 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. The common terms are (2x-1) and (3x^2+1). (dy)/(dx) = (2x - 1)(3x^2 + 1) [4(3x^2 + 1) + 12x(2x - 1)] Step 6: Expand and simplify the terms inside the square brackets. (dy)/(dx) = (2x - 1)(3x^2 + 1) [12x^2 + 4 + 24x^2 - 12x] (dy)/(dx) = (2x - 1)(3x^2 + 1) [36x^2 - 12x + 4] Step 7: Factor out 4 from the bracketed term. (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). Drop the next question! 📸