Derivative equation

The idea of derivative equation is quite simple: The LIMIT of the SLOPE.

The slope is equal to change in Y / change in X.
So for a point a, we IMAGINE we have another near point which lies on the SAME LINE with a,
and since we have TWO POINTS now,
we can then let their Y-value Change divided by their X-value Change to get the slope.

There’re two equations for calculating derivative at a point, and the only different thing is how to express the IMAGINARY POINT with respect to the point a, it could either be x or a+h :

or:

How to calculate derivative

Strategy:

• Determine if it’s CONTINUOUS at this point, by:
• See if the point is defined in the interval
• Calculate LIMITS of both RIGHT SIDE and LEFT SIDE of the point.
• If two sides’ limits are the same, then it’s continuous. Otherwise it’s discontinuous.
• Determine if it’s DIFFERENTIABLE (Actually is the process of getting its derivative):
• Apply Derivative equation to get both RIGHT SIDE LIMIT and LEFT SIDE LIMIT.
• If two sides’ limits are the same, then that value is the Derivative at the point. If not, then it’s NOT DIFFERENTIABLE.

Example

Solve:

• See that the point 3 is defined in the interval.
• Left side limit of the point, is using the first equation, and gets the lim g(x) = -7
• Right side limit of the point, is using the second equation, and gets the lim g(x) = -7
• Limits of both sides are the SAME, so it’s continuous, and let’s see if it’s differentiable.
• Apply the derivative equation for both Left side & Right side:

• Both sides’ limits exists but not that same, so it’s not differentiable.

Example

Solve:

• See that the point -1 is defined in the interval.
• Left side limit of the point, is using the first equation, and gets the lim g(x) = 1
• Right side limit of the point, is using the second equation, and gets the lim g(x) = 4
• Limits of both sides are NOT SAME, so it’s not continuous, then of course not differentiable.