Question

Assertion :Derivative of $\frac{x^n-a^n}{x-a}$ for some constant $n$ is $\frac{(n-1)x^n-na{x}^{n-1}+a^n}{(x-a{)}^2}$ Reason: $\frac{d}{x}\left(\frac{u}{v}\right)=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$
where $u$ and $v$ are two distinct functions.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

By division rule (as stated in Reason), we have, $\frac{d}{x}\left(\frac{u}{v}\right)=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$
$\therefore \frac{d}{dx}\left(\frac{x^n-a^n}{x-a}\right)=\frac{\left[\frac{d}{dx}\right(x^n-a^n\left)\right](x-a)-(x^n-a^n)\frac{d}{dx}(x-a)}{(x-a{)}^2}$
$=\frac{n{x}^{n-1}(x-a)-(x^n-a^n)\cdot 1}{(x-a{)}^2}=\frac{n{x}^n-n\cdot x^{n-1}a-x^n+a^n}{(x-a{)}^2}$
$\frac{(n-1)x^n-na{x}^{n-1}+a^n}{(x-a{)}^2}$

Hence, both assertion and reason are correct and reason is the correct explanation for assertion.