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.
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Morning — let's get this done. Problem 1: Find the derivative of y = sqrt((x^2 + 1)/(x^2 - 1)). We can rewrite the function as y = ((x^2 + 1)/(x^2 - 1))^1/2. We use the chain rule: (dy)/(dx) = (1)/(2)u^-1/2 (du)/(dx), where u = (x^2 + 1)/(x^2 - 1). Step 1: Find (du)/(dx) using the quotient rule. Let p = x^2 + 1 and q = x^2 - 1. Then p' = 2x and q' = 2x. (du)/(dx) = (p'q - pq')/(q^2) = ((2x)(x^2 - 1) - (x^2 + 1)(2x))/((x^2 - 1)^2) (du)/(dx) = (2x^3 - 2x - (2x^3 + 2x))/((x^2 - 1)^2) (du)/(dx) = (2x^3 - 2x - 2x^3 - 2x)/((x^2 - 1)^2) (du)/(dx) = (-4x)/((x^2 - 1)^2) Step 2: Apply the chain rule. (dy)/(dx) = (1)/(2)((x^2 + 1)/(x^2 - 1))^-1/2 · ((-4x)/((x^2 - 1)^2)) (dy)/(dx) = (1)/(2)((x^2 - 1)/(x^2 + 1))^1/2 · ((-4x)/((x^2 - 1)^2)) (dy)/(dx) = (1)/(2) sqrt(x^2 - 1)sqrt(x^2 + 1) · (-4x)/((x^2 - 1)^2) (dy)/(dx) = -2x sqrt(x^2 - 1)sqrt(x^2 + 1) (x^2 - 1)^2 Simplify the powers of (x^2 - 1): (x^2 - 1)^2 = (x^2 - 1)^3/2 (x^2 - 1)^1/2. (dy)/(dx) = (-2x)/(sqrt(x^2 + 1) (x^2 - 1)^3/2) The derivative is: (dy)/(dx) = (-2x)/((x^2 - 1)^3/2)sqrt(x^2 + 1) Problem 2: Find the derivative of f(x) = ((e^x)/(1 + e^x)). Step 1: Simplify the function using logarithm properties. 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) Combine the terms: f'(x) = (1 + e^x - e^x)/(1 + e^x) f'(x) = (1)/(1 + e^x) The derivative is: f'(x) = (1)/(1 + e^x) Problem 3: Find the derivative of y = x^ x. We use logarithmic differentiation. Step 1: Take the natural logarithm of both sides. y = (x^ x) Use the logarithm property (a^b) = b a: y = ( x)( x) y = ( x)^2 Step 2: Differentiate both sides with respect to x implicitly. (1)/(y) (dy)/(dx) = 2( x) · (d)/(dx)( x) (1)/(y) (dy)/(dx) = 2( x) · (1)/(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) Using the exponent rule (a^m)/(a^n) = a^m-n: (dy)/(dx) = 2x^ x - 1 x The derivative is: (dy)/(dx) = 2x^ x - 1 x Problem 4: Find the derivative of y = ( x)^ x. We use logarithmic differentiation. Step 1: Take the natural logarithm of both sides. y = (( x)^ x) Use the logarithm property (a^b) = b a: y = ( x)(( x)) Step 2: Differentiate both sides with respect to x implicitly. Use the product rule on the right side. (1)/(y) (dy)/(dx) = (d)/(dx)( x) · ( x) + x · (d)/(dx)(( x)) We know (d)/(dx)( x) = - x. For (d)/(dx)(( x)), use the chain rule: (1)/( x) · (d)/(dx)( x) = (1)/( x) · x = x. So, (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) + (^2 x)/( x) ) The derivative is: (dy)/(dx) = ( x)^ x ( (^2 x)/( x) - x ( x) )