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
Morning ThegirlGodishelping🥰💙🫠 — let's get this done. Here are the derivatives for each function: 1. y = sqrt((x^2 + 1)/(x^2 - 1)) Step 1: Rewrite the function using exponents. y = ((x^2 + 1)/(x^2 - 1))^1/2 Step 2: Apply the chain rule. Let u = (x^2 + 1)/(x^2 - 1). Then y = u^1/2, so (dy)/(dx) = (1)/(2)u^-1/2 · (du)/(dx). (dy)/(dx) = (1)/(2)((x^2 + 1)/(x^2 - 1))^-1/2 · (d)/(dx)((x^2 + 1)/(x^2 - 1)) (dy)/(dx) = (1)/(2)sqrt((x^2 - 1)/(x^2 + 1)) · (d)/(dx)((x^2 + 1)/(x^2 - 1)) Step 3: Use the quotient rule for (d)/(dx)((x^2 + 1)/(x^2 - 1)). Let f = x^2 + 1 and g = x^2 - 1. Then f' = 2x and g' = 2x. (d)/(dx)((f)/(g)) = (f'g - fg')/(g^2) = ((2x)(x^2 - 1) - (x^2 + 1)(2x))/((x^2 - 1)^2) = (2x^3 - 2x - (2x^3 + 2x))/((x^2 - 1)^2) = (2x^3 - 2x - 2x^3 - 2x)/((x^2 - 1)^2) = (-4x)/((x^2 - 1)^2) Step 4: Substitute this back into the chain rule expression. (dy)/(dx) = (1)/(2)sqrt((x^2 - 1)/(x^2 + 1)) · (-4x)/((x^2 - 1)^2) (dy)/(dx) = (-2x)/((x^2 - 1)^2) sqrt((x^2 - 1)/(x^2 + 1)) (dy)/(dx) = (-2x)/((x^2 - 1)^2) sqrt(x^2 - 1)sqrt(x^2 + 1) (dy)/(dx) = (-2x)/((x^2 - 1)^2 - 1/2) sqrt(x^2 + 1) (dy)/(dx) = (-2x)/((x^2 - 1)^3/2) sqrt(x^2 + 1) (dy)/(dx) = (-2x)/((x^2 - 1)sqrt(x^2 - 1)x^2 + 1) --- 2. f(x) = ((e^x)/(1 + e^x)) Step 1: Use logarithm properties to simplify the function. f(x) = (e^x) - (1 + e^x) f(x) = x - (1 + e^x) Step 2: Differentiate f(x) with respect to x. f'(x) = (d)/(dx)(x) - (d)/(dx)((1 + e^x)) f'(x) = 1 - (1)/(1 + e^x) · (d)/(dx)(1 + e^x) f'(x) = 1 - (1)/(1 + e^x) · (e^x) f'(x) = 1 - (e^x)/(1 + e^x) Step 3: Combine the terms. f'(x) = (1 + e^x)/(1 + e^x) - (e^x)/(1 + e^x) f'(x) = (1 + e^x - e^x)/(1 + e^x) f'(x) = (1)/(1 + e^x) --- 3. y = x^ x Step 1: Use logarithmic differentiation. Take the natural logarithm of both sides. y = (x^ x) y = ( x)( x) y = ( x)^2 Step 2: Differentiate both sides with respect to x. (1)/(y) (dy)/(dx) = 2( x) · (d)/(dx)( x) (1)/(y) (dy)/(dx) = 2( x) · (1)/(x) (1)/(y) (dy)/(dx) = (2 x)/(x) Step 3: Solve for (dy)/(dx). (dy)/(dx) = y · (2 x)/(x) Substitute y = x^ x back into the equation. (dy)/(dx) = x^ x · (2 x)/(x) --- 4. y = ( x)^ x Step 1: Use logarithmic differentiation. Take the natural logarithm of both sides. y = (( x)^ x) y = ( x)(( x)) 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) + x · (d)/(dx)(( x)) (1)/(y) (dy)/(dx) = (- x) ( x) + x · ((1)/( x) · x) (1)/(y) (dy)/(dx) = - x ( x) + (^2 x)/( x) (1)/(y) (dy)/(dx) = - x ( x) + x x Step 3: Solve for (dy)/(dx). (dy)/(dx) = y (- x ( x) + x x) Substitute y = ( x)^ x back into the equation. (dy)/(dx) = ( x)^ x ( x x - x ( x)) --- 5. x^3 + y^3 = 3axy 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 3axy. (d)/(dx)(x^3) + (d)/(dx)(y^3) = (d)/(dx)(3axy) 3x^2 + 3y^2 (dy)/(dx) = 3a ( (d)/(dx)(x) · y + x · (d)/(dx)(y) ) 3x^2 + 3y^2 (dy)/(dx) = 3a ( 1 · y + x · (dy)/(dx) ) 3x^2 + 3y^2 (dy)/(dx) = 3ay + 3ax (dy)/(dx) Step 2: Rearrange the equation to group terms with (dy)/(dx) on one side and other terms on the other side. 3y^2 (dy)/(dx) - 3ax (dy)/(dx) = 3ay - 3x^2 Step 3: Factor out (dy)/(dx). (dy)/(dx)(3y^2 - 3ax) = 3ay - 3x^2 Step 4: Solve for (dy)/(dx). (dy)/(dx) = (3ay - 3x^2)/(3y^2 - 3ax) Factor out 3 from the numerator and denominator. (dy)/(dx) = (ay - x^2)/(y^2 - ax) Got more? Send 'em!