2 Differentiating a quotient

In this Section we consider functions of the form $y=\frac{f\left(x\right)}{g\left(x\right)}$ . To find the derivative of such a function we make use of the following Key Point:

Example 10

Find the derivative of $y=\frac{lnx}{x}$

Solution

Here $f\left(x\right)=lnx$ and $g\left(x\right)=x$

$\therefore \frac{df}{dx}=\frac{1}{x}$ and $\frac{dg}{dx}=1$

Hence $\frac{dy}{dx}=\frac{x\left(\frac{1}{x}\right)-1\left(lnx\right)}{{\left[x\right]}^2}=\frac{1-lnx}{x^2}$

Task!

Obtain the derivative of $y=\frac{sinx}{x^2}$

1. using the formula for differentiating a product and
2. using the formula for differentiating a quotient.

1. Write $y=x^{-2}sinx$ then use the product rule to find $\frac{dy}{dx}$ :

   Answer

   $y=x^{-2}sinx\therefore \frac{dy}{dx}=\left(-2x^{-3}\right)sinx+x^{-2}cosx$ $\therefore \frac{dy}{dx}=\frac{-2sinx+xcosx}{x^3}$

2. Now use the quotient rule instead to find $\frac{dy}{dx}$ :

   Answer

   $y=\frac{sinx}{x^2}\therefore \frac{dy}{dx}=\frac{x^2\left(cosx\right)-\left(2x\right)sinx}{{\left(x^2\right)}^2}=\frac{xcosx-2sinx}{x^3}$

Exercise

Find the derivatives of the following:

1. $\left(2x^3-4x^2\right)\left(3x^5+x^2\right)$
2. $\frac{2x^3+4}{x^2-4x+1}$
3. $\frac{x^2+2x+1}{x^2-2x+1}$
4. $\left(x^2+3\right)\left(2x-5\right)\left(3x+2\right)$
5. $\frac{\left(2x+1\right)\left(3x-1\right)}{x+5}$
6. $\left(lnx\right)sinx$
7. $\left(lnx\right)∕sinx$
8. $e^x∕x^2$
9. $\frac{e^{x}sinx}{cos2x}$

Answer