Question

The derivative of $\frac{sinx}{x}$ will be equal to .

• A. $\frac{cosx}{x}$
• B. $\frac{xcosx-sinx}{x^2}$
• C. $-\frac{sinx}{x^2}$
• D. $\frac{7x(8e^x-4x{e}^x-1)}{(4e^x+x{)}^2}$

Answer 1

The correct option is B $\frac{xcosx-sinx}{x^2}$

The function is the quotient of two functions $sinxandx$. So we will use the quotient rule which says

$\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{f^{\prime }\left(x\right)\times g\left(x\right)-g^{\prime }\left(x\right)\times f\left(x\right)}{\left(g\right(x){)}^2}$

$wheref\left(x\right)=sinx,f^{\prime }\left(x\right)=cosxg\left(x\right)=x,g^{\prime }\left(x\right)=1$
Substituing in the quotient rule formula,
$\frac{d}{dx}\left(\frac{sinx}{x}\right)=\frac{cosx.x-1.sinx}{x^2}=\frac{xcosx-sinx}{x^2}$