Derivative of Sum and Difference of Two or More Functions

Differentiation is a fundamental concept in calculus that deals with the rate at which a function changes. When working with the sum or difference of two or more functions, the rules of differentiation allow us to find the derivative of the combined function in a straightforward manner.

Basic Rules

The derivative of a function measures how the function's output changes as its input changes. The rules for finding the derivative of a sum or difference of functions are based on the linearity of the derivative operator. These rules are:

1. Sum Rule: The derivative of the sum of two functions is the sum of the derivatives of those functions.
2. Difference Rule: The derivative of the difference of two functions is the difference of the derivatives of those functions.

Mathematically, these rules can be expressed as follows:

• Sum Rule: If $f(x)$ and $g(x)$ are differentiable functions, then $(f + g)'(x) = f'(x) + g'(x)$.
• Difference Rule: If $f(x)$ and $g(x)$ are differentiable functions, then $(f - g)'(x) = f'(x) - g'(x)$.

Table of Differences and Important Points

Property	Sum Rule	Difference Rule
Formula	$(f + g)'(x) = f'(x) + g'(x)$	$(f - g)'(x) = f'(x) - g'(x)$
Linearity	Yes	Yes
Commutativity	Yes	No (Subtraction is not commutative)
Associativity	Yes	Yes (With respect to subtraction)
Usage	When adding functions	When subtracting functions

Examples

Let's illustrate these rules with examples.

Example 1: Sum Rule

Consider two functions $f(x) = x^2$ and $g(x) = 3x$. To find the derivative of the sum $h(x) = f(x) + g(x)$, we apply the sum rule.

$$ h(x) = f(x) + g(x) = x^2 + 3x $$

Using the sum rule:

$$ h'(x) = (x^2 + 3x)' = (x^2)' + (3x)' = 2x + 3 $$

Example 2: Difference Rule

Now consider the difference of the same functions $f(x) = x^2$ and $g(x) = 3x$. To find the derivative of the difference $h(x) = f(x) - g(x)$, we apply the difference rule.

$$ h(x) = f(x) - g(x) = x^2 - 3x $$

Using the difference rule:

$$ h'(x) = (x^2 - 3x)' = (x^2)' - (3x)' = 2x - 3 $$

Example 3: Multiple Functions

If we have more than two functions, the sum and difference rules can be applied repeatedly. For instance, for three functions $f(x)$, $g(x)$, and $h(x)$:

$$ (k(x))' = (f + g + h)'(x) = f'(x) + g'(x) + h'(x) $$

And for the difference:

$$ (k(x))' = (f - g - h)'(x) = f'(x) - g'(x) - h'(x) $$

Example 4: Combining Rules

We can also combine the sum and difference rules. For example:

$$ (k(x))' = (f + g - h)'(x) = f'(x) + g'(x) - h'(x) $$

Conclusion

The sum and difference rules are powerful tools in calculus that simplify the process of differentiating complex functions that are composed of simpler functions added or subtracted together. By understanding and applying these rules, one can efficiently find the derivatives of such combined functions. Remember that these rules are a direct consequence of the linearity of the derivative operator, which allows us to treat each function separately when differentiating.