Question:
Find the derivative of \(\frac{x^n-a^n}{x-a}\) for some constant a.
Find the derivative of \(\frac{x^n-a^n}{x-a}\) for some constant a.
Updated On: Oct 25, 2023
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Solution and Explanation
Let f (x)= \(\frac{x^n-a^n}{x-a}\)
\(\Rightarrow\)f'(x)= \(\frac{d}{dx}\)(\(\frac{x^n-a^n}{x-a}\))
By the quotient rule,
f'(x)= (x-a)\(\frac{d}{dx}\)( xn -an) - (xn -an)\(\frac{d}{dx}\)\(\frac{(x-a)}{(x-a)^2}\)
= \(\frac{(x-a)(nx^{n-1}-0)-(x^n -a^n)}{(x-a)^2}\)
= \(\frac{nx^n-a^nx^{n-a}-x^n+a^n}{(x-a)^2}\)
\(\Rightarrow\)f'(x)= \(\frac{d}{dx}\)(\(\frac{x^n-a^n}{x-a}\))
By the quotient rule,
f'(x)= (x-a)\(\frac{d}{dx}\)( xn -an) - (xn -an)\(\frac{d}{dx}\)\(\frac{(x-a)}{(x-a)^2}\)
= \(\frac{(x-a)(nx^{n-1}-0)-(x^n -a^n)}{(x-a)^2}\)
= \(\frac{nx^n-a^nx^{n-a}-x^n+a^n}{(x-a)^2}\)
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