## Differentiating Sums of Functions

Many functions are built by summing two simpler functions. For example, is the sum of the identity function and the constant function We will verify the

Sum Rule

Differentiating sums is pretty easy. Let's consider the average rate of change of the function over the interval to

This shows that the average rate of change of the sum is just the sum of the average rates of change of and

As becomes very small, the two average rates of change are very close to the derivatives and This shows us that

Example:
 Consider the function We know that the derivative that is, the derivative is equal to 1 at every point. This is to be expected since the graph of this function is the straight line through with slope one. Again, the tangent line at any point will be the same graph and so the derivative will be 1 at any point.