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

Asked by Pragya Singh | 1 year ago |  52

Solution :-

Let f(x)​ $$x^n+ax^{n-1}+a^2x^{n-2}+....+a^{n-1}x+a^n$$

$$\dfrac{d}{dx}f(x)= ( x^n+ax^{n-1}+a^2x^{n-2}+....+a^{n-1}x+a^n)$$

$$\dfrac{d}{dx}(x^n)+a\dfrac{d}{dx}(x^{n-1})+a^2+\dfrac{d}{dx}(x^{n-2})$$

$$+....+a^{n-1}\dfrac{d}{dx}(x)+a^n\dfrac{d}{dx}(1)$$

On using theorem $$\dfrac{d}{dx}(x^n)=nx^{n-1}$$ we obtain

$$f'(x)=nx^{n-1}+a(n-1)x^{n-2}+a^2(n-2)x^{n-3}$$

$$+...+a^{n-1}+a^n(0)$$

$$f'(x)=nx^{n-1}+a(n-1)x^{n-2}+a^2(n-2)x^{n-3}$$

$$+...+a^{n-1}+a^n$$

Answered by Abhisek | 1 year ago

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