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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Alright — let's do this. 1. To evaluate _x (x^2 - x)/(x^3 + 2x) using L'Hôpital's rule: Step 1: Check the form of the limit. As x , both the numerator (x^2 - x) and the denominator (x^3 + 2x) approach . This is an indeterminate form ()/(), so L'Hôpital's rule can be applied. Step 2: Differentiate the numerator and the denominator. (d)/(dx)(x^2 - x) = 2x - 1 (d)/(dx)(x^3 + 2x) = 3x^2 + 2 Step 3: Apply L'Hôpital's rule. _x (x^2 - x)/(x^3 + 2x) = _x (2x - 1)/(3x^2 + 2) Again, as x , this is an indeterminate form ()/(). Apply L'Hôpital's rule again. Step 4: Differentiate the new numerator and denominator. (d)/(dx)(2x - 1) = 2 (d)/(dx)(3x^2 + 2) = 6x Step 5: Apply L'Hôpital's rule a second time. _x (2x - 1)/(3x^2 + 2) = _x (2)/(6x) Step 6: Evaluate the limit. As x , 6x , so (2)/(6x) 0. _x (2)/(6x) = 0 The final answer is 0. 2. Differentiate the following functions: a) y = 3^2x Step 1: Rewrite the function using exponents. y = (1+3x)^2/3x Step 2: Apply the quotient rule, (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2). Let u = (1+3x)^2/3 and v = x. Step 3: Find u' using the chain rule. u' = (2)/(3)(1+3x)^(2/3)-1 · (d)/(dx)(1+3x) u' = (2)/(3)(1+3x)^-1/3 · 3 u' = 2(1+3x)^-1/3 Step 4: Find v'. v' = (d)/(dx)(x) = 1 Step 5: Substitute u, v, u', v' into the quotient rule formula. (dy)/(dx) = 2(1+3x)^-1/3 · x - (1+3x)^2/3 · 1x^2 Step 6: Simplify the expression. Factor out (1+3x)^-1/3 from the numerator. (dy)/(dx) = (1+3x)^-1/3 [2x - (1+3x)^2/3(1+3x)^1/3]x^2 (dy)/(dx) = (1+3x)^-1/3 [2x - (1+3x)]x^2 (dy)/(dx) = (1+3x)^-1/3 [2x - 1 - 3x]x^2 (dy)/(dx) = (1+3x)^-1/3 [-x - 1]x^2 (dy)/(dx) = (-(x+1))/(x^2 (1+3x)^1/3) The final answer is -(x+1)/(x^2 [3]1+3x). b) y = ((3)/(4)x^2) Step 1: Apply the chain rule. The derivative of (u) is -(u)(u) · u'. Let u = (3)/(4)x^2. Step 2: Find u'. u' = (d)/(dx)((3)/(4)x^2) = (3)/(4) · 2x = (3)/(2)x Step 3: Substitute u and u' into the chain rule formula. (dy)/(dx) = -((3)/(4)x^2)((3)/(4)x^2) · ((3)/(2)x) Step 4: Rearrange the terms. (dy)/(dx) = -(3)/(2)x ((3)/(4)x^2)((3)/(4)x^2) The final answer is -(3)/(2)x ((3)/(4)x^2)((3)/(4)x^2). c) x^2y + y^2 + 4x = 1 Step 1: Differentiate both sides of the equation with respect to x. Remember to use implicit differentiation for terms involving y. Step 2: Differentiate x^2y using the product rule. (d)/(dx)(x^2y) = (d)/(dx)(x^2) · y + x^2 · (d)/(dx)(y) = 2xy + x^2(dy)/(dx) Step 3: Differentiate y^2 using the chain rule. (d)/(dx)(y^2) = 2y(dy)/(dx) Step 4: Differentiate 4x. (d)/(dx)(4x) = 4 Step 5: Differentiate the constant 1. (d)/(dx)(1) = 0 Step 6: Combine the differentiated terms. 2xy + x^2(dy)/(dx) + 2y(dy)/(dx) + 4 = 0 Step 7: Isolate terms containing (dy)/(dx) on one side. x^2(dy)/(dx) + 2y(dy)/(dx) = -2xy - 4 Step 8: Factor out (dy)/(dx). (dy)/(dx)(x^2 + 2y) = -2xy - 4 Step 9: Solve for (dy)/(dx). (dy)/(dx) = (-2xy - 4)/(x^2 + 2y) (dy)/(dx) = -(2(xy + 2))/(x^2 + 2y) The final answer is -(2(xy + 2))/(x^2 + 2y). Send me the next one 📸