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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Welcome back Kheny’s — missed you this week. 1. Evaluate _x (x^2 - x)/(x^3 + 2x) using L'Hôpital's rule. Step 1: Check the form of the limit. As x , the numerator x^2 - x and the denominator x^3 + 2x . This is an indeterminate form ()/(), so L'Hôpital's rule can be applied. Step 2: Apply L'Hôpital's rule by differentiating the numerator and the denominator. _x (d)/(dx)(x^2 - x)(d)/(dx)(x^3 + 2x) = _x (2x - 1)/(3x^2 + 2) Step 3: Check the form of the new limit. As x , the numerator 2x - 1 and the denominator 3x^2 + 2 . This is still an indeterminate form ()/(), so apply L'Hôpital's rule again. Step 4: Apply L'Hôpital's rule a second time. _x (d)/(dx)(2x - 1)(d)/(dx)(3x^2 + 2) = _x (2)/(6x) Step 5: Evaluate the limit. _x (2)/(6x) = _x (1)/(3x) = 0 The limit is 0. 2. Differentiate the following functions: a) y = 3^2 Step 1: Rewrite the function using fractional exponents. y = (1+3x^2)^2/3 Step 2: Apply the chain rule (dy)/(dx) = n u^n-1 (du)/(dx), where u = 1+3x^2 and n = (2)/(3). (dy)/(dx) = (2)/(3)(1+3x^2)^(2)/(3)-1 · (d)/(dx)(1+3x^2) Step 3: Differentiate the inner function. (d)/(dx)(1+3x^2) = 0 + 3(2x) = 6x Step 4: Substitute and simplify. (dy)/(dx) = (2)/(3)(1+3x^2)^-1/3 · (6x) (dy)/(dx) = (12x)/(3)(1+3x^2)^-1/3 (dy)/(dx) = 4x(1+3x^2)^-1/3 The derivative is 4x(1+3x^2)^-1/3 or (4x)/([3]1+3x^2). b) y = cosec (3)/(4)x^2 Step 1: Recall the derivative of cosec(u), which is -cosec(u)cot(u)(du)/(dx). Here, u = (3)/(4)x^2. Step 2: Differentiate u with respect to x. (du)/(dx) = (d)/(dx)((3)/(4)x^2) = (3)/(4)(2x) = (3)/(2)x Step 3: Apply the chain rule. (dy)/(dx) = -cosec((3)/(4)x^2)cot((3)/(4)x^2) · ((3)/(2)x) Step 4: Rearrange the terms. (dy)/(dx) = -(3)/(2)x cosec((3)/(4)x^2)cot((3)/(4)x^2) The derivative is -(3)/(2)x cosec((3)/(4)x^2)cot((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 the product rule for x^2y and the chain rule for y^2. (d)/(dx)(x^2y) + (d)/(dx)(y^2) + (d)/(dx)(4x) = (d)/(dx)(1) Step 2: Apply the differentiation rules. For (d)/(dx)(x^2y): Use the product rule (u'v + uv'), where u=x^2 and v=y. (2x)y + x^2(dy)/(dx) For (d)/(dx)(y^2): Use the chain rule. 2y(dy)/(dx) For (d)/(dx)(4x): 4 For (d)/(dx)(1): 0 Step 3: Substitute these derivatives back into the equation. 2xy + x^2(dy)/(dx) + 2y(dy)/(dx) + 4 = 0 Step 4: Group terms containing (dy)/(dx) on one side and other terms on the other side. x^2(dy)/(dx) + 2y(dy)/(dx) = -2xy - 4 Step 5: Factor out (dy)/(dx). (dy)/(dx)(x^2 + 2y) = -2xy - 4 Step 6: Solve for (dy)/(dx). (dy)/(dx) = (-2xy - 4)/(x^2 + 2y) This can also be written as: (dy)/(dx) = -(2(xy + 2))/(x^2 + 2y) The derivative is -(2(xy + 2))/(x^2 + 2y). Got more? Send 'em!