Recipe

Build a derivative from the definition

Use when: you're asked for f'(x) (or f' at a specific point) and no shortcut rule is available yet

Goal: a simplified expression for f'(x), or its exact value at one point, with the limit already taken

Taught in The definition of the derivative

1. 1

   Substitute x+h for every x in the formula for f.

   Example

   For f(x)=x^2-4x: f(x+h)=(x+h)^2-4(x+h).

   Why: the difference quotient needs the function's value one step away from x, and this is the only way to get it without a shortcut rule.

2. 2

   Compute f(x+h)-f(x), then expand and collect like terms.

   Example

   \big[(x+h)^2-4(x+h)\big]-\big[x^2-4x\big] = x^2+2xh+h^2-4x-4h-x^2+4x = 2xh+h^2-4h.

   Why: this is the numerator of the difference quotient, and expanding it fully is what exposes the shared factor of h the next step needs.

3. 3

   Every remaining term should carry a factor of h; factor it out and cancel.

   Example

   \dfrac{2xh+h^2-4h}{h}=\dfrac{h(2x+h-4)}{h}=2x+h-4 for h\neq0.

   Why: if a term is left over with no h factor, either the expansion or the subtraction in Step 2 has an error. This is the built-in check.

4. 4

   Take the limit of the simplified expression as h shrinks to 0.

   • a specific point x=a was asked for — substitute x=a first, then let h\to0 (or do both at once).
   • the general derivative function f'(x) was asked for — let h\to0 while keeping x as a variable.

   Why: the simplified form from Step 3 survives substituting h=0 without dividing by zero, which is the entire reason Steps 2–3 existed.