Question

Assertion :Derivative of $(px+q)(\frac{r}{x}+s)$ is $ps+\frac{qr}{x^2}$ Reason: $\frac{d}{dx}\left(uv\right)=u^{\prime }v+u{v}^{\prime }$
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 D Assertion is incorrect but Reason is correct

Let $f\left(x\right)=(px+q)(\frac{r}{x}+s)$
We have, $\frac{d\left(uv\right)}{dx}=u^{\prime }v+u{v}^{\prime }$
$\therefore f^{\prime }\left(x\right)=\left[\frac{d}{dx}\right(px+q\left)\right](\frac{r}{x}+s)+(px+q)\frac{d}{dx}(\frac{r}{x}+s)$
$=p(\frac{r}{x}+s)+(px+q)\left(\frac{-r}{x^2}\right)$
$=\frac{pr}{x}+ps-\frac{pr}{x}-\frac{qr}{x^2}=ps-\frac{qr}{x^2}$

Hence, assertion is incorrect but reason is correct.