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Question

Mathematics Question on Derivatives

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

Answer

Let f (x)= xn−anx−a\frac{x^n-a^n}{x-a}
⇒\Rightarrowf'(x)= ddx\frac{d}{dx}(xn−anx−a\frac{x^n-a^n}{x-a})
By the quotient rule,
f'(x)= (x-a)ddx\frac{d}{dx}( xn -an) - (xn -an)\frac{d}{dx}$$\frac{(x-a)}{(x-a)^2}
= (x−a)(nxn−1−0)−(xn−an)(x−a)2\frac{(x-a)(nx^{n-1}-0)-(x^n -a^n)}{(x-a)^2}
= nxn−anxn−a−xn+an(x−a)2\frac{nx^n-a^nx^{n-a}-x^n+a^n}{(x-a)^2}