Change and Slope
Slope compares change in output to change in input.
For a straight line:
If x increases by 2 and y increases by 6, the slope is:
This means: for each one unit of input change, the output changes by 3 units
on average over that interval.
Average Change
The formula above gives average slope between two points:
It answers a question over an interval. If a loss drops from 10 to 4 while a
weight moves from 1 to 3, the average slope is:
The negative sign matters. It says the output decreased as the input increased over that interval.
Local Slope
Curves do not have one slope everywhere.
At each point, a curve can have a different local slope. The derivative measures that local slope.
This is the first bridge to learning. A loss curve may go up in one place and down in another. The derivative tells what is happening near the current point.
Average slope is about two points. Local slope is about one point and the instantaneous direction of change there.
This distinction matters in training. An average loss change over a large move can hide what happened along the way. A derivative tries to answer the smaller question: if we move a tiny amount from the current point, which way does the loss initially go?
An input changes from x = 1 to x = 3. The output changes from y = 4 to
y = 10.
What is the average slope?
Compute it first, then check your number.
Hint
Compute change in y divided by change in x.
Solution
Average slope is output change divided by input change.
The output change is:
10 - 4 = 6
The input change is:
3 - 1 = 2
So the average slope is 6 / 2 = 3.
A value changes from y = 10 to y = 4 while x changes from 1 to 3.
What is the average slope?
Compute it first, then check your number.
Hint
Compute (4 - 10) / (3 - 1).
Solution
Average slope keeps the sign of the output change.
The output change is:
4 - 10 = -6
The input change is:
3 - 1 = 2
So the average slope is -6 / 2 = -3. The negative sign says the output fell
as the input increased.
Does average slope use one point or two points?
Answer it first, then check.
Hint
Average slope compares an output change across an input interval.
Solution
Average slope uses two points. It compares how the output changed between them.
If the average slope over an interval is 0, did the output change overall
between the two endpoints?
Answer it first, then check.
Hint
Zero slope means Delta y = 0 over that interval.
Solution
No. A zero average slope means the two endpoint outputs are the same, so the
overall endpoint change is 0.
Enter 1 if an average slope of 0 between two endpoints can still hide
nonzero local slopes between those endpoints.
Compute it first, then check your number.
Hint
Imagine a curve that goes up, then comes back down to the same endpoint value.
Solution
Enter 1. Average slope only compares the two endpoints. The function may have
changed in between, even if the endpoint change is zero.
Before Moving On
Slope is rate of change. Derivatives are local slopes.