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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1. y = x^(1)/(2) - 3x^(3)/(4) Step 1: Power rule (d)/(dx) x^n = n x^n-1. (d)/(dx) ( x^(1)/(2) ) = (1)/(2) x^(1)/(2)-1 = (1)/(2) x^-(1)/(2) Step 2: (d)/(dx) ( -3x^(3)/(4) ) = -3 · (3)/(4) x^(3)/(4)-1 = -(9)/(4) x^-(1)/(4) Step 3: Sum the derivatives. y' = (1)/(2) x^-(1)/(2) - (9)/(4) x^-(1)/(4) y' = (1)/(2sqrt(x)) - (9)/(4 x^1/4) 2. y = x - x - 2 x Step 1: (d)/(dx) x = 1, (d)/(dx) (- x) = x, (d)/(dx) (-2 x) = -2 x. Step 2: Sum the derivatives. y' = 1 + x - 2 x y' = 1 + x - 2 x 3. y = e^x x Step 1: Product rule (d)/(dx) (uv) = u'v + uv', where u = e^x, v = x. u' = e^x, v' = ^2 x Step 2: Substitute. y' = e^x x + e^x ^2 x Step 3: Factor. y' = e^x ( x + ^2 x) y' = e^x ( x + ^2 x) 4. y = ( x)^2 - 4 x Step 1: Chain rule for ( x)^2: let u = x, (dy)/(du) = 2u, (du)/(dx) = (1)/(x). (d)/(dx) ( x)^2 = 2 x · (1)/(x) Step 2: (d)/(dx) (-4 x) = -4 · (1)/(x). Step 3: Sum. y' = (2 x)/(x) - (4)/(x) = (2 x - 4)/(x) y' = (2 x - 4)/(x) 5. y = (3x + 1) Step 1: Chain rule (d)/(dx) u = u · u', u = 3x + 1, u' = 3. y' = (3x + 1) · 3 = 3 (3x + 1) y' = 3(3x + 1) 6. y = ( x) Step 1: Chain rule (d)/(dx) u = (1)/(u) u', u = x, u' = x x. y' = (1)/( x) · x x Step 2: Simplify. y' = x y' = x 7. y = (x^2 + 1)^5 Step 1: Chain rule (d)/(dx) u^5 = 5u^4 u', u = x^2 + 1, u' = 2x. y' = 5(x^2 + 1)^4 · 2x = 10x (x^2 + 1)^4 y' = 10x(x^2 + 1)^4 8. y = (1)/(x^2 + 1) Step 1: Quotient rule (d)/(dx) (u)/(v) = (u'v - uv')/(v^2), u=1, v = x^2 + 1. u' = 0, v' = 2x Step 2: Substitute. y' = (0 · (x^2 + 1) - 1 · 2x)/((x^2 + 1)^2) = (-2x)/((x^2 + 1)^2) y' = -(2x)/((x^2) + 1)^2 9. y = [f(x)]^-1 - h(x) Step 1: Chain rule for [f(x)]^-1: (d)/(dx) u^-1 = -u^-2 u', u = f(x), u' = f'(x). (d)/(dx) [f(x)]^-1 = -[f(x)]^-2 f'(x) = -(f'(x))/([f(x)]^2) Step 2: (d)/(dx) [-h(x)] = -h'(x). Step 3: Sum. y' = -(f'(x))/([f(x)]^2) - h'(x) y' = -(f'(x))/([f(x)]^2) - h'(x)