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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You're on a roll — let's tackle these differentiation problems. Problem 6: Differentiate y = (sqrt((1-x)/(1+x))) with respect to x. We can simplify this function using a trigonometric substitution. Step 1: Let x = (2). Then dx = -2(2)\,d. Substitute x into the expression inside the square root: sqrt((1-x)/(1+x)) = sqrt((1-(2))/(1+(2))) Step 2: Use the double angle identities 1-(2) = 2^2 and 1+(2) = 2^2. sqrt((2^2)/(2^2)) = sqrt(^2) = || Assuming x (-1, 1), then 2 (0, ), which means (0, /2). In this interval, > 0. So, sqrt((1-x)/(1+x)) = . Step 3: Substitute this back into the original function. y = () y = Step 4: Express in terms of x. Since x = (2), we have 2 = x, so = (1)/(2) x. y = (1)/(2) x Step 5: Differentiate y with respect to x. (dy)/(dx) = (d)/(dx)((1)/(2) x) (dy)/(dx) = (1)/(2) · (-(1)/(sqrt(1-x^2))) (dy)/(dx) = -(1)/(2sqrt(1-x^2)) Problem 7: Find f'(x) for f(x) = e^3x (5x^2 - 2x + 1). We use the product rule: (d)/(dx)(uv) = u'v + uv'. Let u = e^3x and v = (5x^2 - 2x + 1). Step 1: Find the derivative of u. u' = (d)/(dx)(e^3x) = e^3x · (d)/(dx)(3x) = 3e^3x Step 2: Find the derivative of v. v' = (d)/(dx)((5x^2 - 2x + 1)) = (1)/(5x^2 - 2x + 1) · (d)/(dx)(5x^2 - 2x + 1) = (10x - 2)/(5x^2 - 2x + 1) Step 3: Apply the product rule. f'(x) = u'v + uv' f'(x) = (3e^3x)(5x^2 - 2x + 1) + (e^3x)((10x - 2)/(5x^2 - 2x + 1)) Step 4: Factor out e^3x. f'(x) = e^3x(3(5x^2 - 2x + 1) + (10x - 2)/(5x^2 - 2x + 1)) Problem 8: Find (dy)/(dx) if y = ((4x^2))/((x+1)^3). We use the quotient rule: (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2). Let u = (4x^2) and v = (x+1)^3. Step 1: Find the derivative of u. u' = (d)/(dx)((4x^2)) = (4x^2) · (d)/(dx)(4x^2) = (4x^2) · 8x = 8x (4x^2) Step 2: Find the derivative of v. v' = (d)/(dx)((x+1)^3) = 3(x+1)^2 · (d)/(dx)(x+1) = 3(x+1)^2 · 1 = 3(x+1)^2 Step 3: Apply the quotient rule. (dy)/(dx) = ((8x (4x^2))(x+1)^3 - ((4x^2))(3(x+1)^2))/(((x+1)^3)^2) (dy)/(dx) = ((8x (4x^2))(x+1)^3 - 3(4x^2)(x+1)^2)/((x+1)^6) Step 4: Factor out (x+1)^2 from the numerator. (dy)/(dx) = ((x+1)^2 [8x (4x^2)(x+1) - 3(4x^2)])/((x+1)^6) (dy)/(dx) = (8x(x+1)(4x^2) - 3(4x^2))/((x+1)^4) (dy)/(dx) = (8x^2 (4x^2) + 8x (4x^2) - 3(4x^2))/((x+1)^4) Problem 9: Find (dy)/(dx) if 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) (1)/(y) (dy)/(dx) = (1) x + x((1)/(x)) (1)/(y) (dy)/(dx) = x + 1 Step 3: Solve for (dy)/(dx). (dy)/(dx) = y( x + 1) Substitute y = x^x back into the equation: (dy)/(dx) = x^x( x + 1) Problem 10: Differentiate y = ( x + x)^n with respect to x. Step 1: Use the logarithm property (A^n) = n A to simplify the function. y = n ( x + x) Step 2: Differentiate y with respect to x using the chain rule. (dy)/(dx) = n · (1)/( x + x) · (d)/(dx)( x + x) Step 3: Find the derivative of ( x + x). Recall that (d)/(dx)( x) = x x and (d)/(dx)( x) = ^2 x. (d)/(dx)( x + x) = x x + ^2 x Factor out x: x x + ^2 x = x ( x + x) Step 4: Substitute this back into the derivative expression. (dy)/(dx) = n · (1)/( x + x) · x ( x + x) The term ( x + x) cancels out. (dy)/(dx) = n x Got more? Send 'em!