Quotient Rule for differentiation
Quotient Rule:
Rule:
 For two differentiable functions $u(x)$ and $v(x)$ where $v(x)\mathrm{\ne}0$, the derivative of their quotient is: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right]=\frac{v(x)\cdot {u}^{\mathrm{\prime}}(x)u(x)\cdot {v}^{\mathrm{\prime}}(x)}{[v(x){]}^{2}}$.
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 $f(x)=\frac{{x}^{2}}{\mathrm{sin}(x)}$
 Let $u(x)={x}^{2}$ and $v(x)=\mathrm{sin}(x)$
 ${u}^{\mathrm{\prime}}(x)=2x$ and ${v}^{\mathrm{\prime}}(x)=\mathrm{cos}(x)$
 Apply the quotient rule: $\frac{d}{dx}\left[\frac{{x}^{2}}{\mathrm{sin}(x)}\right]=\frac{\mathrm{sin}(x)\cdot 2x{x}^{2}\cdot \mathrm{cos}(x)}{[\mathrm{sin}(x){]}^{2}}$
Properties and Notes:

Applicability and Conditions:
 The quotient rule is applicable when differentiating a quotient of two functions, and the denominator function is not zero.

Avoiding Ambiguity:
 The rule helps avoid the ambiguity that arises when directly differentiating fractions using the power rule.

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.