Discrimination between left and right limits of derivatives and left and right derivatives

Preliminary knowledge

When calculating the derivative , Often for a single special point , We'll use derivatives Definition To calculate . And calculation Within the interval The derivative of , Is to use common Derivative formula Calculate .

understand

And the reason for that is ,
In the same interval is a formula .
and The formulas of left and right derivatives contain special points , May involve Two different formulas , therefore , For special points, we'd better use Definition Come and ask for .
Simply speaking ： Special points are defined by , Use formula in the interval .

Problem solving

First , We know that the left and right limits of the derivative are not the value of this point , Left limit , Right limit , Function value , Is three values . therefore , The left limit belongs to the left interval , The right limit belongs to the right interval , Function values are special points .
therefore , We directly launch , seek The formula is used when the left and right limits （ The interval does not contain special points ）, seek The left derivative and the right derivative are defined （ The formula contains special points ）

Example show

Saying without practicing the fake trick

For example, in this question , except x=0 It's special , All other places are sections .

Pay attention to the second question and the third question The second question is the left and right derivatives Definition is used The third question is the left-right limit Using the formula , It can be found that it fits our inference .