Sum and Difference rules for differentiation
Sum Rule:
Rule:
 The derivative of the sum of two functions is the sum of their derivatives: $\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}[f(x)]+\frac{d}{dx}[g(x)]$.
Example:
 $f(x)={x}^{2}+3x$
 ${f}^{\mathrm{\prime}}(x)=\frac{d}{dx}[{x}^{2}]+\frac{d}{dx}[3x]=2x+3$
Difference Rule:
Rule:
 The derivative of the difference of two functions is the difference of their derivatives: $\frac{d}{dx}[f(x)g(x)]=\frac{d}{dx}[f(x)]\frac{d}{dx}[g(x)]$.
Example:
 $g(x)=\mathrm{sin}(x){x}^{3}$
 ${g}^{\mathrm{\prime}}(x)=\frac{d}{dx}[\mathrm{sin}(x)]\frac{d}{dx}[{x}^{3}]=\mathrm{cos}(x)3{x}^{2}$
Properties of Sum and Difference Rules:

Applicability:
 The rules apply to functions composed of sums or differences of elementary functions, constants, or more complex functions.

Linearity:
 These rules exhibit linearity, allowing for the derivative of each term in the sum or difference to be evaluated separately.

Extensibility:
 They can be extended to more than two functions by repeatedly applying the rules to each pair of functions.
Application of Sum and Difference Rules:
 Simplification of Derivatives:
 Used to break down complex functions into simpler components for easier differentiation.
 Functions with Multiple Terms:
 Useful in finding derivatives of polynomials, trigonometric functions, and algebraic expressions composed of multiple terms.