What are the sum and difference rules for derivatives?

Sum and Difference Rules for Derivatives

In calculus, the sum and difference rules are essential for finding derivatives of functions that are sums or differences of other functions. Here's how they work:

1. Sum Rule: The derivative of the sum of two functions is the sum of their derivatives. In other words, if you have two functions, f(x) and g(x), then:

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

This means that to find the derivative of a sum of functions, you simply find the derivative of each function separately and then add them together.

1. Difference Rule: Similarly, the derivative of the difference of two functions is the difference of their derivatives. That is:

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

To find the derivative of a difference of functions, you find the derivative of each function separately and then subtract the derivative of the second function from the first.

Here are a few examples to illustrate these rules:

• If $f(x) = 3x^2$ and $g(x) = 2x$, then:

  • $(f(x) + g(x))' = (3x^2 + 2x)' = 6x + 2$
  • $(f(x) - g(x))' = (3x^2 - 2x)' = 6x - 2$

• If $f(x) = \sin(x)$ and $g(x) = \cos(x)$, then:

  • $(f(x) + g(x))' = (\sin(x) + \cos(x))' = \cos(x) - \sin(x)$
  • $(f(x) - g(x))' = (\sin(x) - \cos(x))' = \cos(x) + \sin(x)$