Question

Find the derivative of xn + ax n-1 + a2xn-2+……...+a n-1x + an for some fixed real number a.

Answer 1

Let f'(x): = xn + ax n-1 + a2xn-2+……...+a n-1x + an
∴f(x)=$\frac{d}{dx}$(xn + ax n-1 + a2xn-2+……...+a n-1x + an)
=$\frac{d}{dx}$(xn) + a$\frac{d}{dx}$(xn-1) + a2$\frac{d}{dx}$(xn-2) + …..... + an-1$\frac{d}{dx}$(x) + an $\frac{d}{dx}$(1)
On using theorem $\frac{d}{dx}$xn
= nxn-1, we obtain f'(x)=nxn-1 +a(n-1)xn-2 + a2 (n-2)xn-3 +...+an-1+an (0)
=nxn-1+ a(n-1)xn-2+a2(n-2)xn-3 +...+an-1