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NCERT Class XI Mathematics - Limits and Derivatives - Solutions

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Question : 40 of 72
Marks: +1, -0
Find the derivative of xn−anx−a\frac{x^n - a^n}{x - a} for some constant a.
Solution:  
Let f (x) = xn−anx−a\frac{x^n - a^n}{x - a} ........(i), where a is a constant.
Differentiating (i) with respect to x, we get
f' (x) =
(x−a)(xn−an)′−(xn−an)⋅(x−a)′(x−a)2\frac{(x-a)(x^n - a^n)' - (x^n - a^n) \cdot (x-a)'}{(x-a)^2}
⇒ f' (x) =
(x−a)(nxn−1−0)−(xn−an)⋅(1)(x−a)2\frac{(x-a)(nx^{n-1} - 0) - (x^n - a^n) \cdot (1)}{(x-a)^2}
⇒ f' (x) = nxn−1(x−a)−xn+an(x−a)2\frac{nx^{n-1}(x-a) - x^n + a^n}{(x-a)^2}
= nxn−anxn−1−xn+an(x−a)2\frac{nx^n - anx^{n-1} - x^n + a^n}{(x-a)^2}
∴ f' (x) = nxn−anxn−1−xn+an(x−a)2\frac{nx^n - anx^{n-1} - x^n + a^n}{(x-a)^2}
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