Find the derivative of xn-an-x-a for some constant a-

Here f(x) = x^n a^n/x a ∴ f'(x) =d/dx[x^n a^n/x a] = (x a)d/dx(x^n a^n) (x^n a^n)d/dx(x a)/(x a)^2 = (x a)× nx^n 1 (x^n a^n) × 1/(x a)^2 = nx^2 an x^n 1 x^n+a^n ...

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Byju's Answer

Standard XI

Mathematics

Derivative from First Principle

Find the deri...

Question

Find the derivative of $\frac{x^{n} - a^{n}}{x - a}$ for some constant a.

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Solution

Here f(x) = $\frac{x^{n} - a^{n}}{x - a}$

$∴$ f'(x) = $\frac{d}{d x} [\frac{x^{n} - a^{n}}{x - a}]$

= $\frac{(x - a) \frac{d}{d x} (x^{n} - a^{n}) - (x^{n} - a^{n}) \frac{d}{d x} (x - a)}{(x - a)^{2}}$

= $\frac{(x - a) \times n x^{n - 1} - (x^{n} - a^{n}) \times 1}{(x - a)^{2}}$

= $\frac{n x^{2} - a n x^{n - 1} - x^{n} + a^{n}}{(x - a)^{2}}$

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Similar questions

Q. For any constant

a

, find the derivative of

\frac{x^{n} - a^{n}}{x - a}

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

Q. For any constant real number

a

, find the derivative of:

x^{n} + a x^{n - 1} + a^{2} x^{n - 2} + . . . + a^{n - 1} x + a^{n}

Q. Assertion :Derivative of

\frac{x^{n} - a^{n}}{x - a}

for some constant n is

\frac{(n - 1) x^{n} - n a x^{n - 1} + a^{n}}{(x - a)^{2}}

Reason:

\frac{d}{d x} (\frac{u}{v}) = \frac{u^{'} v - u v^{'}}{v^{2}}

where u and v are two distinct functions.

Q. Assertion :Derivative of

\frac{x^{n} - a^{n}}{x - a}

for some constant

n

\frac{(n - 1) x^{n} - n a x^{n - 1} + a^{n}}{(x - a)^{2}}

Reason:

\frac{d}{x} (\frac{u}{v}) = \frac{u^{'} v - u v^{'}}{v^{2}}

where

u

and

v

are two distinct functions.

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Find the derivative of xn−anx−a for some constant a.

Here f(x) = xn−anx−a ∴ f'(x) =ddx[xn−anx−a] = (x−a)ddx(xn−an)−(xn−an)ddx(x−a)(x−a)2 = (x−a)×nxn−1−(xn−an)×1(x−a)2 = nx2−anxn−1−xn+an(x−a)2

Find the derivative of $\frac{x^{n} - a^{n}}{x - a}$ for some constant a.