Sum rule

Sum Rule in mathematics, specifically in the field of calculus, is a fundamental rule used for finding the derivative of the sum of two or more functions. It states that the derivative of a sum of functions is equal to the sum of the derivatives of those functions. This rule is a cornerstone in differential calculus and is instrumental in simplifying the process of differentiation when dealing with complex functions.

Definition

Given two functions, \(f(x)\) and \(g(x)\), the sum rule can be formally expressed as:

\[ \frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x) \]

This means that if you need to find the derivative of the sum of two functions, you can simply find the derivatives of each function individually and then add them together.

Application

The sum rule is widely used in various fields of mathematics and its applications, including physics, engineering, and economics, where differentiation plays a key role in analyzing and solving problems. It is particularly useful in simplifying the process of finding derivatives for complex expressions by breaking them down into simpler parts.

Examples

1. Given \(f(x) = x^2\) and \(g(x) = 3x\), find the derivative of \(f(x) + g(x)\).

Using the sum rule:

\[ \frac{d}{dx}(x^2 + 3x) = \frac{d}{dx}x^2 + \frac{d}{dx}3x = 2x + 3 \]

2. For \(f(x) = \sin(x)\) and \(g(x) = \cos(x)\), find the derivative of \(f(x) + g(x)\).

Applying the sum rule:

\[ \frac{d}{dx}(\sin(x) + \cos(x)) = \frac{d}{dx}\sin(x) + \frac{d}{dx}\cos(x) = \cos(x) - \sin(x) \]

See Also

• Product Rule
• Quotient Rule
• Chain Rule
• Differential Calculus