Find the derivative of x^n-a^n x-a for some constant a. - Vidyadip

Question

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

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Answer

Let $\text{f}(\text{x})=\frac{\text{x}^\text{n}-\text{a}^\text{n}}{\text{x}-\text{a}}$ $\Rightarrow\text{f}'(\text{x})=\frac{\text{d}}{\text{dx}}\bigg(\frac{\text{x}^\text{n}-\text{a}^\text{n}}{\text{x}-\text{a}}\bigg)$ By quotient rule, $ \text{f}'(\text{x})=\frac{({\text{x}-\text{a})\frac{\text{d}}{\text{dx}}(\text{x}^\text{n}-\text{a}^\text{n})-(\text{x}^\text{n}-\text{a}^\text{n})}\frac{\text{d}}{\text{dx}}(\text{x}-\text{a})}{(\text{x}-\text{a})^2}$ $ =\frac{({\text{x}-\text{a})(\text{nx}^\text{n-1}-0)-(\text{x}^\text{n}-\text{a}^\text{n})}}{(\text{x}-\text{a})^2}$ $ =\frac{{\text{nx}^\text{n}-\text{anx}^\text{n-1}-\text{x}^\text{n}+\text{a}^\text{n}}}{(\text{x}-\text{a})^2}$

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