Question

Find the derivative of `x^n  + ax^(n-1) + a^2 x^(n-2) + ...+ a^(n -1) x + a^n` for some fixed real number a.

Answer 1

Let `f(x) = xⁿ + ax^{n - 1} + a²x^{n - 2} + ... + a^{n - 1} + aⁿ

∴ f(x) = `d/(dx) (x^n + ax^(n - 1) + a^2 x^(n - 2) + ... + a^(n - 1) + a^n)`

= `d/(dx) (x^n) + ad/(dx)(x^(n - 1)) + a^2 d/(dx)(x^(n - 2)) + ... + a^(n - 1)d/(dx)(x) + a^n d/(dx) (1)`

On using theorem `d/(dx)x^n = nx^(n - 1)` we obtain

`f'("x") = "nx"^("n" - 1) + "a". ("n" - 1)"x"^("n" - 2) + "a"^2 . ("n" - 2)"x"^("n" - 3) + ...... + "a"^("n" - 1). 1`

= `"nx"^("n" - 1) + ("n" - 1)"ax"^("n" - 2) + ("n" - 2) "a"^2"x"^("n" - 3) + ....... + "a"^("n" - 1)`