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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To find the derivative of y = (2x^2 + 1)/(x^5 + 2x), we will use the quotient rule, which states that if y = (u)/(v), then (dy)/(dx) = (u'v - uv')/(v^2). Step 1: Identify u and v and find their derivatives. Let u = 2x^2 + 1. Then u' = (d)/(dx)(2x^2 + 1) = 4x. Let v = x^5 + 2x. Then v' = (d)/(dx)(x^5 + 2x) = 5x^4 + 2. Step 2: Apply the quotient rule formula. (dy)/(dx) = (u'v - uv')/(v^2) Substitute the expressions for u, v, u', and v': (dy)/(dx) = ((4x)(x^5 + 2x) - (2x^2 + 1)(5x^4 + 2))/((x^5 + 2x)^2) Step 3: Expand the terms in the numerator. First term: (4x)(x^5 + 2x) = 4x · x^5 + 4x · 2x = 4x^6 + 8x^2. Second term: (2x^2 + 1)(5x^4 + 2) = 2x^2(5x^4) + 2x^2(2) + 1(5x^4) + 1(2) = 10x^6 + 4x^2 + 5x^4 + 2. Step 4: Substitute the expanded terms back into the numerator and simplify. (dy)/(dx) = ((4x^6 + 8x^2) - (10x^6 + 5x^4 + 4x^2 + 2))/((x^5 + 2x)^2) Distribute the negative sign in the numerator: (dy)/(dx) = (4x^6 + 8x^2 - 10x^6 - 5x^4 - 4x^2 - 2)/((x^5 + 2x)^2) Combine like terms in the numerator: (dy)/(dx) = ((4x^6 - 10x^6) - 5x^4 + (8x^2 - 4x^2) - 2)/((x^5 + 2x)^2) (dy)/(dx) = (-6x^6 - 5x^4 + 4x^2 - 2)/((x^5 + 2x)^2) The derivative of the given function is: (dy)/(dx) = (-6x^6 - 5x^4 + 4x^2 - 2)/((x^5 + 2x)^2) Send me the next one 📸