# Quotient Rule for differentiation

### Quotient Rule:

Rule:

• For two differentiable functions $u\left(x\right)$ and $v\left(x\right)$ where $v\left(x\right)\mathrm{\ne }0$, the derivative of their quotient is: $\frac{d}{dx}\left[\frac{u\left(x\right)}{v\left(x\right)}\right]=\frac{v\left(x\right)\cdot {u}^{\mathrm{\prime }}\left(x\right)-u\left(x\right)\cdot {v}^{\mathrm{\prime }}\left(x\right)}{\left[v\left(x\right){\right]}^{2}}$.

Explanation:

• The quotient rule states that when differentiating the quotient of two functions, the derivative is found using a specific formula involving the derivatives of the numerator and denominator.

Example:

• Consider $f\left(x\right)=\frac{{x}^{2}}{\mathrm{sin}\left(x\right)}$
• Let $u\left(x\right)={x}^{2}$ and $v\left(x\right)=\mathrm{sin}\left(x\right)$
• ${u}^{\mathrm{\prime }}\left(x\right)=2x$ and ${v}^{\mathrm{\prime }}\left(x\right)=\mathrm{cos}\left(x\right)$
• Apply the quotient rule: $\frac{d}{dx}\left[\frac{{x}^{2}}{\mathrm{sin}\left(x\right)}\right]=\frac{\mathrm{sin}\left(x\right)\cdot 2x-{x}^{2}\cdot \mathrm{cos}\left(x\right)}{\left[\mathrm{sin}\left(x\right){\right]}^{2}}$

### Properties and Notes:

1. Applicability and Conditions:

• The quotient rule is applicable when differentiating a quotient of two functions, and the denominator function is not zero.
2. Avoiding Ambiguity:

• The rule helps avoid the ambiguity that arises when directly differentiating fractions using the power rule.
3. Handling Complex Ratios:

• Useful in finding derivatives of complex algebraic expressions or functions expressed as ratios.

### Applications of the Quotient Rule:

• Rate of Change Problems:

• Used to analyze rates of change in various contexts, such as motion problems in physics or growth and decay problems in biology or economics.
• Optimization and Critical Points:

• Helpful in optimization problems where finding critical points or extreme values of functions is necessary.