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The standard rule (smaller denominator implies larger fraction) applies strictly when the common numerator is positive. If the common numerator is zero, all such fractions are equal to zero, resulting in equality.

Conditions: Comparing fractions $\frac{n}{a}$ and $\frac{n}{b}$ with $0 < a < b$; Case 1: $n > 0$ (standard case); Case 2: $n = 0$; Case 3: $n < 0$

The standard rule (smaller denominator implies larger fraction) applies strictly when the common numerator is positive. If the common numerator is zero, all such fractions are equal to zero, resulting in equality.

Conditions: Comparing fractions $\frac{n}{a}$ and $\frac{n}{b}$ with $0 < a < b$; Case 1: $n > 0$ (standard case); Case 2: $n = 0$; Case 3: $n < 0$

The formula $\frac{d}{dx}f(x)=6x^2$ is presented as the exact algebraic counterpart to the graphical observations. It matches the standard power-rule pattern for monomials—multiplying by the exponent and reducing the exponent by one—and perfectly fits the symmetric, U-shaped distribution of the seven manually estimated tangent slopes.

Conditions: Valid for $f(x)=2x^3$ on all real $x$.; No proof of the general power rule is provided within the segment itself.

The formula $\frac{d}{dx}f(x)=6x^2$ is presented as the exact algebraic counterpart to the graphical observations. It matches the standard power-rule pattern for monomials—multiplying by the exponent and reducing the exponent by one—and perfectly fits the symmetric, U-shaped distribution of the seven manually estimated tangent slopes.

Conditions: Valid for $f(x)=2x^3$ on all real $x$.; No proof of the general power rule is provided within the segment itself.