Given:
[tex]y=\frac{x^2-a^2}{x-a}[/tex]
To find:
The derivative.
Explanation:
a) Using the quotient rule,
[tex]\begin{gathered} y^{\prime}=\frac{(x-a)\frac{d}{dx}(x^2-a^2)-(x^2-a^2)\frac{d}{dx}(x-a)}{(x-a)^2} \\ =\frac{(x-a)(2x)-(x^2-a^2)}{(x-a)^2} \\ =\frac{(x-a)(2x)-(x-a)(x+a)}{(x-a)^2}\text{ \lbrack Since, }x^2-a^2=(x+a)(x-a)] \\ =\frac{(x-a)[2x-(x+a)]}{(x-a)^2} \\ =\frac{(x-a)(x-a)}{(x-a)^2} \\ y^{\prime}=1 \end{gathered}[/tex]
b)
By expanding the product and simplifying the quotient, we get
[tex]\begin{gathered} y=\frac{x^2-a^2}{x-a} \\ =\frac{(x+a)(x-a)}{(x-a)} \\ y=x+a \end{gathered}[/tex]
Differentiating with respect to x we get,
[tex]y^{\prime}=1[/tex]
The part (a) answer and the part (b) answer are the same.
Hence verified.
Final answer:
The derivative for the given problem is 1.
