Exercise 3: Is -|x| Differentiable at x=0? — Possible Solution
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GIVEN
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f(x) = -|x|, checking differentiability at x=0

STEP 1: RIGHT-SIDE DIFFERENCE QUOTIENT, h=0.01
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(f(0+0.01) - f(0)) / 0.01 = (-|0.01| - (-|0|)) / 0.01
= (-0.01 - 0) / 0.01 = -0.01/0.01 = -1.0

STEP 2: LEFT-SIDE DIFFERENCE QUOTIENT, h=-0.01
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(f(0-0.01) - f(0)) / (-0.01) = (-|-0.01| - 0) / (-0.01)
= (-0.01 - 0) / (-0.01) = -0.01 / -0.01 = 1.0

STEP 3: DO THEY AGREE?
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Right-side value: -1.0
Left-side value: 1.0

They do NOT agree - exactly the opposite sign pattern from this
chapter's own |x| example (which was +1 on the right, -1 on the
left; here it's reversed, -1 on the right, +1 on the left, since
-|x| is just |x| flipped upside down).

RESULT
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Since the right-side and left-side difference quotients disagree,
no single limit exists for the difference quotient at x=0 -
therefore -|x| is NOT differentiable at x=0, for exactly the same
underlying reason |x| itself isn't: both have a sharp corner at
that point rather than a smooth, well-defined tangent line.

WHY THIS WORKS AS AN ANSWER
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Both one-sided difference quotients are computed explicitly using
this chapter's own method, the disagreement is stated plainly
rather than glossed over, and the conclusion (not differentiable)
is drawn directly from that disagreement using the same reasoning
this chapter already established for the closely related |x| case,
rather than treated as a fresh, unconnected result.