Exercise 2: The Chain Rule on (3x+1)^2 — Possible Solution
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STEP 1: IDENTIFY THE OUTER AND INNER FUNCTIONS
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h(x) = (3x+1)^2

Outer function: f(u) = u^2
Inner function: g(x) = 3x+1

h(x) = f(g(x))

STEP 2: APPLY THE CHAIN RULE
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f'(u) = 2u (power rule)
g'(x) = 3 (derivative of 3x+1)

h'(x) = f'(g(x)) . g'(x) = 2(3x+1) . 3 = 6(3x+1)

STEP 3: EVALUATE AT x=1
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h'(1) = 6(3*1+1) = 6*4 = 24

STEP 4: VERIFY NUMERICALLY
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Using central difference numerical differentiation with h=1e-6 at
x=1: the numerical result is approximately 23.999999997 - matching
24 to well within the expected floating-point precision (a tiny,
harmless rounding difference in the 8th decimal place, not a real
discrepancy).

RESULT
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h'(1) = 24, confirmed both by the chain rule and by numerical
differentiation.

WHY THIS WORKS AS AN ANSWER
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The outer and inner functions are identified explicitly before
applying the chain rule (rather than jumping straight to an
answer), and the result is independently cross-checked with
numerical differentiation, with the tiny floating-point discrepancy
correctly recognized as expected precision noise rather than a real
disagreement.