Definition
Sum Rule
Let
be an open interval and let be differentiable functions. Then the sum is differentiable on and its derivative is: In Leibniz notation:
Intuition
Adding slopes
The derivative measures the rate of change of a function. If two functions change simultaneously, the total change is simply the sum of the individual changes. A small step
produces changes and , so: Dividing by
and taking the limit gives the sum rule.