Product Rule for Differentiation
Product Rule:
Rule:
 For two differentiable functions $u(x)$ and $v(x)$, the derivative of their product is: $\frac{d}{dx}[u(x)\cdot v(x)]={u}^{\mathrm{\prime}}(x)\cdot v(x)+u(x)\cdot {v}^{\mathrm{\prime}}(x)$.
Explanation:
 The product rule states that when differentiating the product of two functions, the derivative is found by taking the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.
Example:
 Consider $f(x)={x}^{2}\cdot \mathrm{sin}(x)$
 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 product rule: $\frac{d}{dx}[{x}^{2}\cdot \mathrm{sin}(x)]=2x\cdot \mathrm{sin}(x)+{x}^{2}\cdot \mathrm{cos}(x)$
Properties and Notes:

Linearity of the Rule:
 The product rule follows a linear structure, allowing for the differentiation of products of functions.

Extensibility:
 The rule extends to more than two functions by repeatedly applying the product rule to each pair of functions.

Derivatives of More Complex Functions:
 Useful in finding derivatives of functions where two or more functions are multiplied together, such as polynomial products, trigonometric functions, and algebraic expressions.
Applications of the Product Rule:

Optimization Problems:
 Used in finding critical points where the derivative is zero to determine maximum or minimum values of functions.

Physics and Engineering:
 Applied in problems involving rates of change, such as in kinematics, dynamics, and electrical engineering.