The sum rule states that the derivative of the sum of two functions is the sum of their individual derivatives. Let's prove this with what we know about limits.

Suppose we have a function k(x) that is the sum of two functions f(x) and g(x). The derivative of k(x) is given by the derivative of the sum of the functions f(x) and g(x).

We substitute these values into the equation defining the derivative and rearrange the terms.

Next, we apply the limit law, which states that the limit of a sum is the sum of the limits. We now have two limits involving f(x) and g(x).

These two limits are expressions for the two derivatives f′(x) and g′(x).

Remember that we defined k′(x) as the sum of f′(x) and g′(x). By substituting this value into the equation, we arrive at the sum rule.

The sum rule can be extended to any number of functions.

Now, suppose we are interested in the derivative of the difference of two functions f(x) and g(x). Which of our differentiation rules should we use to solve this problem?

First, let's first rearrange the terms to form a sum, rather than a difference, and then apply the sum rule.

Applying the constant multiple rule yields our desired result.

This equation is known as the difference rule, which states that the derivative of the difference of two functions is the difference of the derivatives of the two functions.

Now, let's look at a different function.

When x equals two, we find that the value of the derivative is zero.

Now, let's work out a problem that requires both the sum and difference rules. What is the derivative of this function? Click the "Submit" button after selecting your answer.

When x is two, what is the value of the derivative of this function? Click the "Submit" button after entering your answer.

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