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Home  >>  CBSE XI  >>  Math  >>  Limits and Derivatives

Find the derivatives of $x^n+ax^{n-1}+a^2x^{n-2}+...+a^{n-1}x+a^n$ for some fixed real number $a$

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Let $f(x) = x^n+ax^{n-1}+a^2x^{n-2}+...+a^{n-1}x+a^n$
$ \therefore f'(x) =\large\frac{d}{dx}$$ (x^n+ax^{n-1}+a^2x^{n-2}+...+a^{n-1}x+a^n)$
$ = \large\frac{d}{dx}$$ (x^n)+ a\large\frac{d}{dx}$$(x^{n-1})+a^2\large\frac{d}{dx}$$(x^{n-2})+...+a^{n-1}\large\frac{d}{dx}$$(x)+a^n\large\frac{d}{dx}$$(1)$
On using theorem $\large\frac{d}{dx}$$x^n =nx^{n-1}$, we obtain
$f'(x) =nx^{n-1}+a(n-1)x^{n-2}+a^2(n-2)x^{n-3}+...a^{n-1}+a^n(0)$
$ =nx^{n-1}+a(n-1)x^{n-2}+a^2(n-2)x^{n-3}+...a^{n-1}$
answered Apr 9, 2014 by thanvigandhi_1