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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Here are the solutions for the differentiation problems: 1) y = 6x The student's answer is incorrect. Step 1: Apply the chain rule. The derivative of (u) is (u) · u'. Here, u = 6x, so u' = 6. (dy)/(dx) = (6x) · 6 Step 2: Simplify the expression. (dy)/(dx) = 6 6x The correct answer is 6 6x. 2) y = 3 x The student's answer is correct. Step 1: Use the constant multiple rule and the derivative of x. The derivative of x is ^2 x. (dy)/(dx) = 3 · (d)/(dx)( x) Step 2: Substitute the derivative. (dy)/(dx) = 3 ^2 x The correct answer is 3 ^2 x. 3) y = (1)/(2)x Step 1: Apply the chain rule. The derivative of (u) is (u) · u'. Here, u = (1)/(2)x, so u' = (1)/(2). (dy)/(dx) = ((1)/(2)x) · (1)/(2) Step 2: Simplify the expression. (dy)/(dx) = (1)/(2) ((1)/(2)x) The answer is (1)/(2) ((1)/(2)x). 4) y = (2x + 5) Step 1: Apply the chain rule. The derivative of (u) is -(u) · u'. Here, u = 2x + 5, so u' = 2. (dy)/(dx) = -(2x + 5) · 2 Step 2: Simplify the expression. (dy)/(dx) = -2 (2x + 5) The answer is -2 (2x + 5). 5) y = (3x - 1) Step 1: Apply the chain rule. The derivative of (u) is ^2(u) · u'. Here, u = 3x - 1, so u' = 3. (dy)/(dx) = ^2(3x - 1) · 3 Step 2: Simplify the expression. (dy)/(dx) = 3 ^2(3x - 1) The answer is 3 ^2(3x - 1). 6) y = 4 2x Step 1: Apply the constant multiple rule and the chain rule. The derivative of (u) is ^2(u) · u'. Here, u = 2x, so u' = 2. (dy)/(dx) = 4 · ^2(2x) · 2 Step 2: Simplify the expression. (dy)/(dx) = 8 ^2(2x) The answer is 8 ^2(2x). 7) y = x - x Step 1: Differentiate each term separately. The derivative of x is x. The derivative of x is - x. (dy)/(dx) = (d)/(dx)( x) - (d)/(dx)( x) Step 2: Substitute the derivatives. (dy)/(dx) = x - (- x) Step 3: Simplify the expression. (dy)/(dx) = x + x The answer is x + x. 8) y = 2 x + 3 x Step 1: Differentiate each term separately using the constant multiple rule. The derivative of 2 x is 2(- x) = -2 x. The derivative of 3 x is 3(^2 x) = 3 ^2 x. (dy)/(dx) = (d)/(dx)(2 x) + (d)/(dx)(3 x) Step 2: Substitute the derivatives. (dy)/(dx) = -2 x + 3 ^2 x The answer is -2 x + 3 ^2 x. 9) y = 2x - 2x Step 1: Differentiate each term using the chain rule. For 2x: u = 2x, u' = 2. Derivative is -(2x) · 2 = -2 2x. For 2x: u = 2x, u' = 2. Derivative is (2x) · 2 = 2 2x. (dy)/(dx) = (-2 2x) - (2 2x) Step 2: Simplify the expression. (dy)/(dx) = -2 2x - 2 2x (dy)/(dx) = -2( 2x + 2x) The answer is -2( 2x + 2x). 10) y = 2x + 4x Step 1: Differentiate each term using the chain rule. For 2x: u = 2x, u' = 2. Derivative is ^2(2x) · 2 = 2 ^2 2x. For 4x: u = 4x, u' = 4. Derivative is (4x) · 4 = 4 4x. (dy)/(dx) = (2 ^2 2x) + (4 4x) The answer is 2 ^2 2x + 4 4x. 11) f(x) = 2x (x)/(2) Step 1: Apply the product rule: (uv)' = u'v + uv'. Let u = 2x and v = (x)/(2). Step 2: Find the derivatives of u and v. u' = (d)/(dx)(2x) = 2. v' = (d)/(dx)( (x)/(2)) = ((x)/(2)) · (1)/(2) = (1)/(2) ((x)/(2)). Step 3: Substitute into the product rule formula. f'(x) = (2)( (x)/(2)) + (2x)((1)/(2) (x)/(2)) Step 4: Simplify the expression. f'(x) = 2 (x)/(2) + x (x)/(2) The answer is 2 (x)/(2) + x (x)/(2). 12) f(x) = ( 2x)/( 3x) Step 1: Apply the quotient rule: ((u)/(v))' = (u'v - uv')/(v^2). Let u = 2x and v = 3x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)( 2x) = (2x) · 2 = 2 2x. v' = (d)/(dx)( 3x) = -(3x) · 3 = -3 3x. Step 3: Substitute into the quotient rule formula. f'(x) = ((2 2x)( 3x) - ( 2x)(-3 3x))/(( 3x)^2) Step 4: Simplify the expression. f'(x) = (2 2x 3x + 3 2x 3x)/(^2 3x) The answer is (2 2x 3x + 3 2x 3x)/(^2 3x). 13) f(x) = ^3 x Step 1: Rewrite the function as f(x) = ( x)^3. Step 2: Apply the chain rule. Let u = x, so u' = x. The derivative of u^3 is 3u^2 · u'. f'(x) = 3( x)^2 · x Step 3: Simplify the expression. f'(x) = 3 ^2 x x The answer is 3 ^2 x x. 14) y = sqrt( 2x) Step 1: Rewrite the function as y = ( 2x)^1/2. Step 2: Apply the chain rule. Let u = 2x. Then u' = -(2x) · 2 = -2 2x. The derivative of u^1/2 is (1)/(2)u^-1/2 · u'. (dy)/(dx) = (1)/(2)( 2x)^-1/2 · (-2 2x) Step 3: Simplify the expression. (dy)/(dx) = (1)/(2sqrt( 2x)) · (-2 2x) (dy)/(dx) = (- 2x)/(sqrt( 2x)) The answer is (- 2x)/(sqrt( 2x)). 15) y = (1)/( x) Step 1: Rewrite the function as y = ( x)^-1. Step 2: Apply the chain rule. Let u = x. Then u' = x. The derivative of u^-1 is -1u^-2 · u'. (dy)/(dx) = -1( x)^-2 · x Step 3: Simplify the expression. (dy)/(dx) = -( x)/(^2 x) This can also be written as - x x. The answer is -( x)/(^2 x). 16) f(x) = (1)/( x) Step 1: Rewrite the function as f(x) = ( x)^-1. Step 2: Apply the chain rule. Let u = x. Then u' = - x. The derivative of u^-1 is -1u^-2 · u'. f'(x) = -1( x)^-2 · (- x) Step 3: Simplify the expression. f'(x) = ( x)/(^2 x) This can also be written as x x. The answer is ( x)/(^2 x). Drop the next question! 📸