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Experience the top AI math solver designed for both students and educators. Swiftly tackle math homework using detailed, step-by-step solutions complemented by lucid explanations.

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Given the function f(x) = (3x^2 − 2x + 1)e^(-x), find the first derivative f'(x).
Prove by induction that the sum of the squares of the first n natural numbers is equal to n/6 * (n+1) * (2n+1).
Determine whether the following set of vectors forms a basis for R^3: v1 = [1, 0, 2], v2 = [2, -1, 3], v3 = [0, 1, 1].
Let G be a group and H a subgroup of G. Show that if [G:H] = 2, then H is a normal subgroup of G.