We derive the constant rule, power rule, and sum rule.

First, we introduce a different notation for the derivative which may be more convenient at times.

### The constant rule

The simplest examples of functions, hence the best place to start our investigation, are constant functions. Recall that derivatives measure the rate of change of a function at a given point. This means the derivative of a constant function is zero. Here are some ways to think about this situation.

- The constant function plots a horizontal line—so the slope of the tangent line is .
- If represents the position of an object with respect to time and is constant, then the object is not moving, so its velocity is zero. Hence .
- If represents the velocity of an object with respect to time and is constant, then the object’s acceleration is zero. Hence .

The examples above lead us to our next theorem. To gain intuition, you should compute the derivative of using the limit definition of the derivative.

### The power rule

Next, let’s examine derivatives of powers of a single variable. To gain intuition, you
should compute the derivative of using the limit definition of the derivative. Before
computing this derivative, we should recall the *Binomial Theorem*.

since every term but the first has a factor of .

Therefore,

Let’s consider several examples. We begin with something basic.

Sometimes, it is not as obvious that one should apply the power rule.

The power rule also applies to radicals once we rewrite them as exponents.

### The sum rule

We want to be able to take derivatives of functions “one piece at a time.” The *sum
rule* allows us to do exactly this. The sum rule says that we can add the
rates of change of two functions to obtain the rate of change of the sum of
both functions. For example, viewing the derivative as the velocity of an
object, the sum rule states that to find the velocity of a person walking on a
moving bus, we add the velocity of the bus and the velocity of the walking
person.

We now have the tools to work some more complicated examples.