# Sum and Difference rules for differentiation

### Sum Rule:

Rule:

• The derivative of the sum of two functions is the sum of their derivatives: $\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]+\frac{d}{dx}\left[g\left(x\right)\right]$.

Example:

• $f\left(x\right)={x}^{2}+3x$
• ${f}^{\mathrm{\prime }}\left(x\right)=\frac{d}{dx}\left[{x}^{2}\right]+\frac{d}{dx}\left[3x\right]=2x+3$

### Difference Rule:

Rule:

• The derivative of the difference of two functions is the difference of their derivatives: $\frac{d}{dx}\left[f\left(x\right)-g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]-\frac{d}{dx}\left[g\left(x\right)\right]$.

Example:

• $g\left(x\right)=\mathrm{sin}\left(x\right)-{x}^{3}$
• ${g}^{\mathrm{\prime }}\left(x\right)=\frac{d}{dx}\left[\mathrm{sin}\left(x\right)\right]-\frac{d}{dx}\left[{x}^{3}\right]=\mathrm{cos}\left(x\right)-3{x}^{2}$

### 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.