Sum and Difference rules for differentiation

Sum Rule:

Rule:

  • The derivative of the sum of two functions is the sum of their derivatives: ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)].

Example:

  • f(x)=x2+3x
    • f(x)=ddx[x2]+ddx[3x]=2x+3

Difference Rule:

Rule:

  • The derivative of the difference of two functions is the difference of their derivatives: ddx[f(x)g(x)]=ddx[f(x)]ddx[g(x)].

Example:

  • g(x)=sin(x)x3
    • g(x)=ddx[sin(x)]ddx[x3]=cos(x)3x2

Properties of Sum and Difference Rules:

  1. Applicability:

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

    • These rules exhibit linearity, allowing for the derivative of each term in the sum or difference to be evaluated separately.
  3. 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.