#### Chapter 7 Differential Calculus

Section 7.3 Calculation Rules

# 7.3.3 Product and Quotient of Functions

##### Product and Quotient Rule 7.3.3
Likewise, the product of functions, i.e. $f:=u·v$ with $f\left(x\right)=\left(u·v\right)\left(x\right):=u\left(x\right)·v\left(x\right)$, is differentiable, and the following product rule applies:

$f\text{'}\left(x\right)=u\text{'}\left(x\right)·v\left(x\right)+u\left(x\right)·v\text{'}\left(x\right) .$

The quotient of functions, i.e. $f:=\frac{u}{v}$ with $f\left(x\right)=\left(\frac{u}{v}\right)\left(x\right):=\frac{u\left(x\right)}{v\left(x\right)}$, is defined and differentiable for all $x$ with $v\left(x\right)\ne 0$, and the following quotient rule applies:

$f\text{'}\left(x\right)=\frac{u\text{'}\left(x\right)·v\left(x\right)-u\left(x\right)·v\text{'}\left(x\right)}{{\left(v\left(x\right)\right)}^{2}} .$

These calculation rules shall be illustrated by means of a few examples.
##### Example 7.3.4
Find the derivative of $f:ℝ\to ℝ$ with $f\left(x\right)={x}^{2}·e{}^{x}$. The product rule can be applied choosing, for example, $u\left(x\right)={x}^{2}$ and $v\left(x\right)=e{}^{x}$. The corresponding derivatives are $u\text{'}\left(x\right)=2x$ and $v\text{'}\left(x\right)=e{}^{x}$. Combining these terms according to the product rule results in the derivative of the function $f$:

$f\text{'}:ℝ\to ℝ , x\to f\text{'}\left(x\right)=2xe{}^{x}+{x}^{2}e{}^{x}=\left({x}^{2}+2x\right)e{}^{x} .$

Next, we investigate the tangent function $g$ with $g\left(x\right)=\mathrm{tan}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$ ($\mathrm{cos}\left(x\right)\ne 0$).In order to use the quotient rule we set $u\left(x\right)=\mathrm{sin}\left(x\right)$ and $v\left(x\right)=\mathrm{cos}\left(x\right)$. The corresponding derivatives are $u\text{'}\left(x\right)=\mathrm{cos}\left(x\right)$ and $v\text{'}\left(x\right)=-\mathrm{sin}\left(x\right)$. Combining these terms and applying the quotient rule results in the derivative of the function $g$:

$g\text{'}\left(x\right)=\frac{\mathrm{cos}\left(x\right)·\mathrm{cos}\left(x\right)-\mathrm{sin}\left(x\right)·\left(-\mathrm{sin}\left(x\right)\right)}{{\mathrm{cos}}^{2}\left(x\right)} .$

This result can be transformed into any of the following expressions:

$g\text{'}\left(x\right)=1+{\left(\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}\right)}^{2}=1+{\mathrm{tan}}^{2}\left(x\right)=\frac{1}{{\mathrm{cos}}^{2}\left(x\right)} .$

For the last transformation, the relation ${\mathrm{sin}}^{2}\left(x\right)+{\mathrm{cos}}^{2}\left(x\right)=1$ was used, which was given in Module 5 (see Section 5.6.2).

##### Exercise 7.3.5
Calculate the derivative of $f:ℝ\to ℝ$ with $f\left(x\right)=\mathrm{sin}\left(x\right)·{x}^{3}$ by factorising the product into two factors, taking the derivatives of each single factor, and finally combining the results according to the product rule.
1. The derivative of the left factor $u\left(x\right)$$=$
is $u\text{'}\left(x\right)$$=$ .
2. The derivative of the right factor $v\left(x\right)$$=$
is $v\text{'}\left(x\right)$$=$ .
3. Thus, applying the product rule to $f$ results in $f\text{'}\left(x\right)$$=$ .

##### Exercise 7.3.6
Calculate the derivative of $f:\text{}\right]0;\infty \left[\text{}\to ℝ$ with $f\left(x\right)=\frac{\mathrm{ln}\left(x\right)}{{x}^{2}}$ by splitting the quotient up into numerator and denominator, taking the derivatives of both, and combining them according to the quotient rule.
1. The derivative of the numerator $u\left(x\right)$$=$
is $u\text{'}\left(x\right)$$=$ .
2. The derivative of the denominator $v\left(x\right)$$=$
is $v\text{'}\left(x\right)$$=$ .
3. Thus, applying the quotient rule to $f$ results in $f\text{'}\left(x\right)$$=$ .