Question

Find the derivative of f(x) = 1 + x + x² + x³ +... + x⁵⁰ at x = 1.

For the function f(x) = x¹⁰⁰/100 + x⁹⁹/99 + ... + x²/2 + x + 1. Prove that f′(1) = 100 f′(0).

Find the derivative of xⁿ + axⁿ⁻¹ + a²xⁿ⁻² + . . . + aⁿ⁻¹x + aⁿ for some fixed real number a.