LESSON 3: Limit of the derivative (without proof)
Warming up: By analyzing the graphs deduce and indicate if the functions are continuous in [𝑎, 𝑏]
and differentiable in [𝑎, 𝑏], If not justify the answer.
STEP1 Reading comprehension
The following theorem is effective in doing many exercises about differentiability.
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Theorem: Limit of the derivative
Let 𝑓(𝑥) be a continuous function at 𝑥0 and assume that there exists a neighbourhood
𝐼 𝑜𝑓 𝑥0 so that 𝑓(𝑥) is differentiable in 𝐼 – 𝑥0 . Assuming that the limits of the derivative
function
lim 𝑓 ′ (𝑥) = 𝑙1 𝑎𝑛𝑑 lim+ 𝑓 ′ (𝑥) = 𝑙2 exist in 𝑹 − {𝒙𝟎 }, then
𝑥→𝑥0 −
𝑥→𝑥0
𝑓′− (𝑥0 ) =
𝑙1 𝑎𝑛𝑑 𝑓′+ (𝑥0 ) = 𝑙2
And if 𝒍𝟏 = 𝒍𝟐 the function 𝑓(𝑥) is differentiable at 𝑥0 and 𝑙1 = 𝑙2 = 𝑓 ′ (𝑥0 ).
NOTE: This theorem implies that the derivative function is not allowed to have removable
discontinuities.
HOW TO APPLY THIS THEOREM
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By “Maths in English” , Francesco Andriulli, Petrini Editori
The function can be written in a case-defined ( or a piecewise defined) form. Read the following
examples and the corresponding strategy to solve them.
Example ( a case defined function)
3
𝑥≤2
√𝑥 + 3
−𝑥 2
1−𝑒
2<𝑥≤0
𝜋
𝑓(𝑥) =
𝑥𝑠𝑖𝑛𝑥 0 < 𝑥 ≤ 2
𝜋
2
{ |𝜋 − 𝑥|
<𝑥
It’s convenient to further expand the absolute value b using its definition
3
√𝑥 + 3
2
1 − 𝑒 −𝑥
𝑥𝑠𝑖𝑛𝑥
𝑓(𝑥) =
𝜋−𝑥
{𝑥 − 𝜋
𝑥≤2
2<𝑥≤0
𝜋
0<𝑥≤
2
𝜋
<𝑥≤𝜋
2
𝜋<𝑥
STRATEGIES:
a) Study the continuity of the function: any point of discontinuity will be a point of nondifferentiability.
Applying: The given function is continuous in each sub-interval, thus we need to
check what happens at the sub-interval boundaries using the limits if necessary.
Let’s try by yourselves.
(x=-2 is a jump discontinuity)
b) Compute the derivative function in each sub-interval, indentify the point of nondifferentiability internal to the sub-intervals,
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3
3√(𝑥 +
3)2
2𝑥𝑒 −𝑥
𝑓′(𝑥) =
2
𝑥<2
2<𝑥<0
𝜋
𝑠𝑖𝑛𝑥 + 𝑥𝑐𝑜𝑠𝑥 0 < 𝑥 <
2
𝜋
−1
<𝑥<𝜋
2
{
1
𝜋<𝑥
Applying: Answer the questions .
1. At what point 𝑥0 is the derivative function
undefined?
2. Which is the value of f’ as x approaches
that 𝑥0 ?
3. Which type of no-differentiability is 𝑥0 ?
c) In each boundary point between two sub-intervals, calculate the limits of the derivative
from left and right. The theorem on the limit of the derivative ensures that, at the boundary
point, the function is differentiable if the two limits are equal.
Applying: Answer the following questions in relation to the function f’(x)
1. What happens at the boundaries points?
2. Which types of no-differentiability points does the function have?
3. Plot the graph of the function 𝑓(𝑥).
Applying: Now it’s your turn! Study the differentiability of the functions at page 379 using the
three steps
1. Study the continuity of the function ( Where the function is not continuous it isn’t
differentiable as well)
2. Compute the derivative function and determine the domain of f’(x). (because if the
derivative function is undefined at 𝑥0 then the function f(x) is not differentiable at 𝑥0 )
3. If the function is case- defined , calculate the limits of the derivative from left and right in
each boundary point between two sub-intervals.
But … what happens when the limit of the derivative function doesn’t
exist at all?
We cannot apply the theorem, in other words we cannot deduce anything
but, according to the examples at pag. 359 of our schoolbook, we need to use
the definition of derivative function and calculate the limit of the difference
quotient.