Derive polynomial gradients from first principles

To find the derivative at x = 2, which approach is valid?

Reduce the positive increment. Read the output change and quotient for f(x) = x² − x at x = 1.

Why does a shrinking output change not force the quotient to approach zero?

The exact definition

A derivative is a limit, not a secant at one arbitrarily small increment. Keep the increment nonzero during algebra. You can derive a general rule and then evaluate it, or work at a fixed input from the start

1. 1Fix one input or keep it general To find f′(a), you may insert a before forming the quotient. To find a rule f′(x), keep x general. Both are valid uses of the same limit definition \(\displaystyle f\prime(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\)
2. 2Replace the whole input Substitute x + h for every x and expand each power. Subtract the entire original rule with brackets

Which expression correctly substitutes x + h into f(x) = x² − 3x + 2?

Exact expression

\(\displaystyle f(x)=x^2-3x+2,\quad f(x+h)-f(x)=A xh+h^2-3h\)

After expanding and subtracting, enter the coefficient of xh

Exact expression

\(\displaystyle \frac{2xh+h^2-3h}{h}=2x+h+B,\quad h\ne0\)

Cancel the common factor h. Enter the constant coefficient B of the simplified quotient

Why is the cancellation valid?

Exact expression

\(\displaystyle \frac{f(x+h)-f(x)}{h}=2x+h-3\)

Take the limit and evaluate the derivative at x = 2.

Keep zero coefficients visible

A constant disappears when the original function is subtracted. Missing powers should still receive coefficient zero when a labelled response table asks for them

1. 1Cancel for nonzero increments Every term in the difference numerator has an increment factor. Factor and cancel it only while the increment is nonzero \(\displaystyle \frac{hQ(x,h)}{h}=Q(x,h),\quad h\ne0\)
2. 2Take the limit after simplifying For the resulting polynomial, let the increment approach zero. Terms containing a positive increment power vanish. The surviving expression is the derivative

Exact expression

\(\displaystyle (x+h)^3=x^3+3x^2h+A xh^2+h^3\)

Enter the coefficient of xh² in the expanded shifted cubic

Which notation expresses the same derivative when y = f(t)?

Bounded challenge extensions

Reciprocal and square-root examples use the same definition with extra algebra. Combine fractions for a reciprocal; multiply by a conjugate for a square root. Respect the domain. The generated root tasks use a positive square input, so the final derivative is rational

1. 1Keep the domain Reciprocal inputs must be nonzero. Root examples here use a positive square input and a sufficiently small nonzero increment

Exact expression

\(\displaystyle \lim_{h\to0}\frac1{\sqrt{4+h}+2}\)

For the square-root challenge, take this exact limit at a = 4.