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Each of the seven adjusted orange controls encodes a discrete pair consisting of an input $x$ and an estimated slope $m$. Once these individual samples are deemed sufficiently accurate by the software, it automatically interpolates them to draw a continuous yellow parabola representing the full derivative function across the displayed domain.

Conditions: Enough accuracy has been achieved in estimating the individual slopes.; The underlying function remains the same displayed $f(x)=2x^3$ throughout.

Each of the seven adjusted orange controls encodes a discrete pair consisting of an input $x$ and an estimated slope $m$. Once these individual samples are deemed sufficiently accurate by the software, it automatically interpolates them to draw a continuous yellow parabola representing the full derivative function across the displayed domain.

Conditions: Enough accuracy has been achieved in estimating the individual slopes.; The underlying function remains the same displayed $f(x)=2x^3$ throughout.

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.