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 ThegirlGodishelping🥰💙🫠 — ready when you are. Here are the derivatives for each function: 1. Find (dy)/(dx) if y = ^4(^2(x^3)). Step 1: Rewrite the function to clearly show the layers of the chain rule. y = ((^2(x^3)))^4 Step 2: Apply the chain rule repeatedly. The outermost function is (·)^4. (dy)/(dx) = 4((^2(x^3)))^3 · (d)/(dx)((^2(x^3))) Step 3: Differentiate the next layer, (·). (dy)/(dx) = 4^3(^2(x^3)) · (-(^2(x^3))) · (d)/(dx)(^2(x^3)) Step 4: Differentiate the next layer, (·)^2. (dy)/(dx) = -4^3(^2(x^3))(^2(x^3)) · 2(x^3) · (d)/(dx)((x^3)) Step 5: Differentiate the next layer, (·). (dy)/(dx) = -8^3(^2(x^3))(^2(x^3))(x^3) · (x^3) · (d)/(dx)(x^3) Step 6: Differentiate the innermost layer, x^3. (dy)/(dx) = -8^3(^2(x^3))(^2(x^3))(x^3)(x^3) · (3x^2) Step 7: Simplify the expression. (dy)/(dx) = -24x^2 ^3(^2(x^3))(^2(x^3))(x^3)(x^3) --- 2. Find f'(x) if f(x) = sqrt(x^2 + x^2 + 1). Step 1: Rewrite the function using exponents. f(x) = (x^2 + (x^2 + 1)^1/2)^1/2 Step 2: Apply the chain rule. The outermost function is (·)^1/2. f'(x) = (1)/(2)(x^2 + (x^2 + 1)^1/2)^-1/2 · (d)/(dx)(x^2 + (x^2 + 1)^1/2) Step 3: Differentiate the term inside the brackets. (d)/(dx)(x^2 + (x^2 + 1)^1/2) = 2x + (1)/(2)(x^2 + 1)^-1/2 · (d)/(dx)(x^2 + 1) = 2x + (1)/(2)(x^2 + 1)^-1/2 · (2x) = 2x + x(x^2 + 1)^-1/2 = 2x + (x)/(sqrt(x^2 + 1)) = 2xsqrt(x^2 + 1) + xsqrt(x^2 + 1) = x(2sqrt(x^2 + 1) + 1)sqrt(x^2 + 1) Step 4: Substitute this back into the expression for f'(x). f'(x) = (1)/(2sqrt(x^2 + x^2 + 1)) · x(2sqrt(x^2 + 1) + 1)sqrt(x^2 + 1) f'(x) = x(2sqrt(x^2 + 1) + 1)2sqrt(x^2 + 1)sqrt(x^2 + x^2 + 1) --- 3. Differentiate y = (x^2 + 1)^ x with respect to x. Step 1: Use logarithmic differentiation. Take the natural logarithm of both sides. y = ((x^2 + 1)^ x) y = x · (x^2 + 1) Step 2: Differentiate both sides with respect to x. Use the product rule on the right side. (1)/(y) (dy)/(dx) = (d)/(dx)( x) · (x^2 + 1) + x · (d)/(dx)((x^2 + 1)) (1)/(y) (dy)/(dx) = ( x) (x^2 + 1) + x · ((1)/(x^2 + 1) · 2x) (1)/(y) (dy)/(dx) = x (x^2 + 1) + (2x x)/(x^2 + 1) Step 3: Solve for (dy)/(dx). (dy)/(dx) = y ( x (x^2 + 1) + (2x x)/(x^2 + 1)) Substitute y = (x^2 + 1)^ x back into the equation. (dy)/(dx) = (x^2 + 1)^ x ( x (x^2 + 1) + (2x x)/(x^2 + 1)) --- 4. Find (dy)/(dx) for the equation (x + y) = y^2 x. Step 1: Differentiate both sides of the equation with respect to x. Remember to use the chain rule for terms involving y and the product rule for y^2 x. (d)/(dx)((x + y)) = (d)/(dx)(y^2 x) (x + y) · (d)/(dx)(x + y) = (d)/(dx)(y^2) x + y^2 (d)/(dx)( x) (x + y) (1 + (dy)/(dx)) = 2y (dy)/(dx) x + y^2 (- x) (x + y) + (x + y) (dy)/(dx) = 2y x (dy)/(dx) - y^2 x Step 2: Rearrange the equation to group terms with (dy)/(dx) on one side. (x + y) (dy)/(dx) - 2y x (dy)/(dx) = -y^2 x - (x + y) Step 3: Factor out (dy)/(dx). (dy)/(dx)((x + y) - 2y x) = -y^2 x - (x + y) Step 4: Solve for (dy)/(dx). (dy)/(dx) = (-y^2 x - (x + y))/((x + y) - 2y x) (dy)/(dx) = (y^2 x + (x + y))/(2y x - (x + y)) --- 5. Find the derivative of y = x^5 [3]x - 1(2x + 3)^4 using logarithmic differentiation. Step 1: Rewrite the function using exponents. y = x^5 (x - 1)^1/3(2x + 3)^4 Step 2: Take the natural logarithm of both sides. y = (x^5 (x - 1)^1/3(2x + 3)^4) Step 3: Use logarithm properties to expand the expression. y = (x^5) + ((x - 1)^1/3) - ((2x + 3)^4) y = 5 x + (1)/(3) (x - 1) - 4 (2x + 3) Step 4: Differentiate both sides with respect to x. (1)/(y) (dy)/(dx) = 5 · (1)/(x) + (1)/(3) · (1)/(x - 1) · (d)/(dx)(x - 1) - 4 · (1)/(2x + 3) · (d)/(dx)(2x + 3) (1)/(y) (dy)/(dx) = (5)/(x) + (1)/(3(x - 1)) · (1) - (4)/(2x + 3) · (2) (1)/(y) (dy)/(dx) = (5)/(x) + (1)/(3(x - 1)) - (8)/(2x + 3) Step 5: Solve for (dy)/(dx). (dy)/(dx) = y ((5)/(x) + (1)/(3(x - 1)) - (8)/(2x + 3)) Substitute y = x^5 [3]x - 1(2x + 3)^4 back into the equation. (dy)/(dx) = x^5 [3]x - 1(2x + 3)^4 ((5)/(x) + (1)/(3(x - 1)) - (8)/(2x + 3)) What's next?