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Mathematical Expression Editor
We determine differentiability at a point
Differentiability
Differentiability. The function is said to be differentiable at the point if
the following limit exists: We denote the limit by , the derivative of at
.
Geometrically, the derivative gives us the slope of a tangent line. Conceptually, it
represents an instantaneous rate of change. In this section, we will explore the
three main reasons that a function is not differentiable at a point. These are
discontinuities, corner points, and vertical tangent lines.
If is differentiable at , then is continuous at , i.e., differentiability implies
continuity.
An immediate consequence of this theorem is that if is not continuous at , then it
cannot be differentiable there.
If is not continuous at then is not differentiable at .
If the graph of a function has a removable, jump or infinite discontinuity, then the
function is not differentiable at the corresponding point.
There are two other common reasons that a function might not be differentiable.
Corner Point The function has a corner point at if the difference quotient has a
jump discontinuity at , i.e., are both finite, but they are different.
example 1 The function has a corner point at . The left-hand limit, whereas the
right-hand limit,
The other common occurrence of a point of non-differentiability is at a vertical
tangent line.
Vertical Tangent Line The function has a vertical tangent line at if the vertical
line is tangent to the graph of at the point .
example 2 The function has a vertical tangent line at .
(problem 1) Below is the graph of . Answer the questions below the graph. You can
zoom in on the graph if you need to. Is differentiable at ?
YesNo
If yes, is positive, negative or zero?
(problem 2) Below is the graph of . Answer the questions below the graph. You can
zoom in on the graph if you need to. Is differentiable at ?
YesNo
If no, why not?
discontinuity at corner point at vertical tangent line at
(problem 3) Below is the graph of . Answer the questions below the graph. You can
zoom in on the graph if you need to. Is differentiable at ?
YesNo
If no, why not?
discontinuity at corner point at vertical tangent line at
(problem 4) Below is the graph of . Answer the questions below the graph. You can
zoom in on the graph if you need to. Is differentiable at ?
YesNo
If yes, is positive, negative or zero?
(problem 5) Below is the graph of . Answer the questions below the graph. You can
zoom in on the graph if you need to. Is differentiable at ?
YesNo
If no, why not?
discontinuity at corner point at vertical tangent line at
(problem 6) Below is the graph of . Answer the questions below the graph. You can
zoom in on the graph if you need to. Is differentiable at ?