Question

Find the derivative of xⁿ + ax ^n-1 + a²x^n-2+……...+a ^n-1x + aⁿ for some fixed real number a.

Answer 1

Let f'(x): = xⁿ + ax ^n-1 + a²x^n-2+……...+a ^n-1x + aⁿ
∴f(x)=\(\frac{d}{dx}\)(xⁿ + ax ^n-1 + a²x^n-2+……...+a ^n-1x + aⁿ)
=\(\frac{d}{dx}\)(xⁿ) + a\(\frac{d}{dx}\)(x^n-1) + a²\(\frac{d}{dx}\)(x^n-2) + …..... + a^n-1\(\frac{d}{dx}\)(x) + aⁿ \(\frac{d}{dx}\)(1) 
On using theorem \(\frac{d}{dx}\)xⁿ 
= nx^n-1, we obtain f'(x)=nx^n-1 +a(n-1)x^n-2 + a² (n-2)x^n-3 +...+a^n-1+aⁿ (0)
=nx^n-1+ a(n-1)x^n-2+a²(n-2)x^n-3 +...+a^n-1