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 differentiations for each function: 1. y = x^3 + x Step 1: Differentiate each term using the power rule and the derivative of x. (dy)/(dx) = (d)/(dx)(x^3) + (d)/(dx)( x) (dy)/(dx) = 3x^3-1 + x (dy)/(dx) = 3x^2 + x The derivative is 3x^2 + x. 2. y = x^2 x Step 1: Apply the product rule, (d)/(dx)(uv) = u'v + uv', where u = x^2 and v = x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(x^2) = 2x v' = (d)/(dx)( x) = - x Step 3: Substitute u, v, u', v' into the product rule formula. (dy)/(dx) = (2x)( x) + (x^2)(- x) (dy)/(dx) = 2x x - x^2 x The derivative is 2x x - x^2 x. 3. y = (x^2 + x)/(x) Step 1: Apply the quotient rule, (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2), where u = x^2 + x and v = x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(x^2 + x) = 2x + x v' = (d)/(dx)(x) = 1 Step 3: Substitute u, v, u', v' into the quotient rule formula. (dy)/(dx) = ((2x + x)(x) - (x^2 + x)(1))/(x^2) Step 4: Simplify the expression. (dy)/(dx) = (2x^2 + x x - x^2 - x)/(x^2) (dy)/(dx) = (x^2 + x x - x)/(x^2) The derivative is (x^2 + x x - x)/(x^2). 4. y = (x^2 + 1)( x) Step 1: Apply the product rule, (d)/(dx)(uv) = u'v + uv', where u = x^2 + 1 and v = x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(x^2 + 1) = 2x v' = (d)/(dx)( x) = x Step 3: Substitute u, v, u', v' into the product rule formula. (dy)/(dx) = (2x)( x) + (x^2 + 1)( x) (dy)/(dx) = 2x x + (x^2 + 1) x The derivative is 2x x + (x^2 + 1) x. 5. y = x^3 x Step 1: Apply the product rule, (d)/(dx)(uv) = u'v + uv', where u = x^3 and v = x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(x^3) = 3x^2 v' = (d)/(dx)( x) = ^2 x Step 3: Substitute u, v, u', v' into the product rule formula. (dy)/(dx) = (3x^2)( x) + (x^3)(^2 x) (dy)/(dx) = 3x^2 x + x^3 ^2 x The derivative is 3x^2 x + x^3 ^2 x. 6. y = ( x)/(x^2) Step 1: Apply the quotient rule, (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2), where u = x and v = x^2. Step 2: Find the derivatives of u and v. u' = (d)/(dx)( x) = - x v' = (d)/(dx)(x^2) = 2x Step 3: Substitute u, v, u', v' into the quotient rule formula. (dy)/(dx) = ((- x)(x^2) - ( x)(2x))/((x^2)^2) Step 4: Simplify the expression. (dy)/(dx) = (-x^2 x - 2x x)/(x^4) (dy)/(dx) = (-x(x x + 2 x))/(x^4) (dy)/(dx) = (-(x x + 2 x))/(x^3) The derivative is (-(x x + 2 x))/(x^3). 7. y = x x + x^2 Step 1: Differentiate each term. For x x, apply the product rule. For u = x and v = x: u' = 1 v' = x (d)/(dx)(x x) = (1)( x) + (x)( x) = x + x x Step 2: Differentiate x^2. (d)/(dx)(x^2) = 2x Step 3: Combine the derivatives. (dy)/(dx) = ( x + x x) + 2x (dy)/(dx) = x + x x + 2x The derivative is x + x x + 2x. 8. y = (x^2 + x) x Step 1: Apply the product rule, (d)/(dx)(uv) = u'v + uv', where u = x^2 + x and v = x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(x^2 + x) = 2x + 1 v' = (d)/(dx)( x) = - x Step 3: Substitute u, v, u', v' into the product rule formula. (dy)/(dx) = (2x + 1)( x) + (x^2 + x)(- x) (dy)/(dx) = (2x + 1) x - (x^2 + x) x The derivative is (2x + 1) x - (x^2 + x) x. 9. y = (x^3)/( x) Step 1: Apply the quotient rule,