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, good to see you again. Here are the derivatives for each function: 1. To differentiate y = ^3((4x)), we use the chain rule. Step 1: Rewrite the function. y = (((4x)))^3 Step 2: Apply the power rule and chain rule. (dy)/(dx) = 3(((4x)))^2 · (d)/(dx)(((4x))) Step 3: Differentiate ((4x)) using the chain rule. (d)/(dx)(((4x))) = ((4x)) · (d)/(dx)((4x)) Step 4: Differentiate (4x) using the chain rule. (d)/(dx)((4x)) = -(4x) · (d)/(dx)(4x) = -(4x) · 4 = -4(4x) Step 5: Combine the results. (dy)/(dx) = 3^2((4x)) · ((4x)) · (-4(4x)) (dy)/(dx) = -12(4x)^2((4x))((4x)) The derivative is: -12(4x)^2((4x))((4x)) 2. To differentiate f(x) = sqrt((x^2 + 1)), we use the chain rule. Step 1: Rewrite the function. f(x) = ((x^2 + 1))^1/2 Step 2: Apply the power rule and chain rule. f'(x) = (1)/(2)((x^2 + 1))^-1/2 · (d)/(dx)((x^2 + 1)) Step 3: Differentiate (x^2 + 1) using the chain rule. (d)/(dx)((x^2 + 1)) = (1)/(x^2 + 1) · (d)/(dx)(x^2 + 1) Step 4: Differentiate x^2 + 1. (d)/(dx)(x^2 + 1) = 2x Step 5: Combine the results. f'(x) = (1)/(2sqrt((x^2 + 1))) · (1)/(x^2 + 1) · 2x f'(x) = (2x)/(2(x^2 + 1)sqrt((x^2 + 1))) f'(x) = (x)/((x^2 + 1)sqrt((x^2 + 1))) The derivative is: (x)/((x^2 + 1)sqrt((x^2 + 1))) 3. To differentiate y = e^(ex^2), we use the chain rule. Step 1: Apply the chain rule for e^u. (dy)/(dx) = e^(ex^2) · (d)/(dx)(ex^2) Step 2: Differentiate ex^2 using the chain rule. (d)/(dx)(ex^2) = e^x^2 · (d)/(dx)(x^2) Step 3: Differentiate x^2. (d)/(dx)(x^2) = 2x Step 4: Combine the results. (dy)/(dx) = e^(ex^2) · e^x^2 · 2x (dy)/(dx) = 2xe^x^2e^(ex^2) The derivative is: 2xe^x^2e^(ex^2) 4. To differentiate y = (x^2 + 5)^4 · (3x - 1)^3, we use the product rule (d)/(dx)(uv) = u'v + uv'. Let u = (x^2 + 5)^4 and v = (3x - 1)^3. Step 1: Find u'. u' = 4(x^2 + 5)^3 · (d)/(dx)(x^2 + 5) = 4(x^2 + 5)^3 · 2x = 8x(x^2 + 5)^3 Step 2: Find v'. v' = 3(3x - 1)^2 · (d)/(dx)(3x - 1) = 3(3x - 1)^2 · 3 = 9(3x - 1)^2 Step 3: Apply the product rule. (dy)/(dx) = u'v + uv' (dy)/(dx) = 8x(x^2 + 5)^3 (3x - 1)^3 + (x^2 + 5)^4 · 9(3x - 1)^2 Step 4: Factor out common terms. (dy)/(dx) = (x^2 + 5)^3 (3x - 1)^2 [8x(3x - 1) + 9(x^2 + 5)] (dy)/(dx) = (x^2 + 5)^3 (3x - 1)^2 [24x^2 - 8x + 9x^2 + 45] (dy)/(dx) = (x^2 + 5)^3 (3x - 1)^2 [33x^2 - 8x + 45] The derivative is: (x^2 + 5)^3 (3x - 1)^2 (33x^2 - 8x + 45) 5. To differentiate g(x) = (2x + 1)^3 · (5x^2), we use the product rule (d)/(dx)(uv) = u'v + uv'. Let u = (2x + 1)^3 and v = (5x^2). Step 1: Find u'. u' = 3(2x + 1)^2 · (d)/(dx)(2x + 1) = 3(2x + 1)^2 · 2 = 6(2x + 1)^2 Step 2: Find v'. v' = (5x^2) · (d)/(dx)(5x^2) = (5x^2) · 10x = 10x(5x^2) Step 3: Apply the product rule. g'(x) = u'v + uv' g'(x) = 6(2x + 1)^2 (5x^2) + (2x + 1)^3 · 10x(5x^2) Step 4: Factor out common terms. g'(x) = (2x + 1)^2 [6(5x^2) + 10x(2x + 1)(5x^2)] g'(x) = (2x + 1)^2 [6(5x^2) + (20x^2 + 10x)(5x^2)] The derivative is: (2x + 1)^2 [6(5x^2) + (20x^2 + 10x)(5x^2)] Send me the next one 📸