Exercise 1: x^4 via the Power Rule and via the Product Rule — Possible Solution
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STEP 1: POWER RULE ON f(x) = x^4
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d/dx[x^4] = 4x^3

At x=2: 4 * 2^3 = 4*8 = 32

STEP 2: PRODUCT RULE ON f(x) = x . x^3 (SAME FUNCTION, WRITTEN AS A PRODUCT)
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Let u(x) = x, v(x) = x^3, so f(x) = u(x)*v(x) = x*x^3 = x^4
(confirming this really is the same function).

u'(x) = 1 (power rule on x^1)
v'(x) = 3x^2 (power rule on x^3)

Product rule: f'(x) = u'(x)v(x) + u(x)v'(x) = 1*x^3 + x*3x^2
= x^3 + 3x^3 = 4x^3

At x=2: 4 * 2^3 = 32

RESULT
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Both methods give exactly 32 at x=2 - the power rule applied
directly, and the product rule applied to the same function
written as a product of two simpler pieces, agree perfectly.

WHY THIS WORKS AS AN ANSWER
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The product rule is applied with both u(x) and v(x) explicitly
identified and their own derivatives computed via the power rule,
rather than skipping straight to the answer, and the two
independently-derived results are compared directly to confirm they
genuinely agree rather than just assuming the product rule "should"
work.