Calculus 3.2

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3.2 Differentiability
Arches National Park
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To be differentiable, a function must be
continuous and smooth.
Is the function
continuous?
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Answer Yes
2
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But by definition, what is a derivative?
A LIMIT
or
This is another reason why we need to keep the
definition of the derivative in mind before
moving on to the short cuts for derivatives.
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Differentiable also means that the left and right
limits of the derivative are equal.
Is the function
Differentiable at x 1?
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But a limit only exists if its left and right
limits are equal.
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Differentiable also means that the left and right
limits of the derivative are equal.
Is the function
Differentiable at x 1?
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Answer No
?
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To be differentiable, a function must be
continuous and smooth.
Is the function
Differentiable at x 1?
Notice that on the left side, the slope is
approaching 3
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While on the right side, the slope is 2
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To be differentiable, a function must be
continuous and smooth.
Is the function
differentiable at x 1?
First of all, is the function continuous?
Yes
Now can we show if it is differentiable at x 1?
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To be differentiable, a function must be
continuous and smooth.
Is the function
differentiable at x 1?
2
2
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To be differentiable, a function must be
continuous and smooth.
Is the function
differentiable at x 1?
Notice that on the left side, the slope is
approaching 2
Therefore, the answer is YES.
While on the right side, the slope is 2
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To be differentiable, a function must be
continuous and smooth.
Derivatives will fail to exist at
corner
cusp
discontinuity
vertical tangent
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Most of the functions we study in calculus will
be differentiable.
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There are two theorems on page 110
Since a function must be continuous to have a
derivative, if it has a derivative then it is
continuous.
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Intermediate Value Theorem for Derivatives
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