38. Find the derivative of xn+axn−1+a2xn−2+⋯+an−1x+an for some fixed real number a.

38. Given, f(x)=xn+axn−1+a2xn−2+?+an−1x+an. We know that, ddx(xx)=nxx−1 So, f′(x)=ddxxx+ddxaxx−1+ddxa2xx−2+?+ddx ax−1x+ddxax=nxn−1+a(n−1)xn−2+a2(n−2)xn−3+?+an−1+0. (?daxdx=adx dxand dadx=0whereaisconstant

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

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Alok Kumar Singh | Contributor-Level 10

7 months ago

38. Given, $f (x) = x^{n} + a x^{n - 1} + a^{2} x^{n - 2} + ? + a^{n - 1} x + a^{n} .$

We know that,

$\frac{d}{d x} (x^{x}) = n x^{x - 1}$

So,

$f^{'} (x) = \frac{d}{d x} x^{x} + \frac{d}{d x} a x^{x - 1} + \frac{d}{d x} a^{2} x^{x - 2} + ? + \frac{d}{d x}$ $a^{x - 1} x + \frac{d}{d x} a^{x} = n x^{n - 1} + a (n - 1) x^{n - 2} + a^{2} (n - 2) x^{n - 3} + ? + a^{n - 1} + 0 .$

$\begin{array}{l} (? \frac{d a x}{d x} = a \frac{d x}{d x} \frac{and d a}{d x} = 0whereaisconstant \end{array}$ $\begin{array}{l} \end{array}$ $\begin{array}{l} \end{array}$

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