The Derivative

Setting Up the Difference Quotient

Compute [f(x+h)-f(x)]/h for f(x)=x² and simplify to 2x+h.

difference-quotient The difference quotient [f(x+h)-f(x)]/h measures the average rate of change of f over an interval of width h. Simplifying it algebraically prepares for taking the limit as h approaches zero.

Example

Build and simplify the difference quotient for a function.

highlighted = computed this step

Step 1 — Set up

Start with f of x equals x squared.

f(x)=x2f(x)= x^{2}

Step 2 — Substitute x plus h

Replace x by x plus h.

f(x+h)=(x+h)2f(x+h)= \hlmath{(x+h)^{2}}

Step 3 — Form the quotient

Build f of x plus h minus f of x over h.

(x+h)2−x2h\frac{ \hlmath{(x+h)^{2}} - \hlmath{x^{2}} }{h}

Step 4 — Expand numerator

Expand the numerator.

2xh+h2\hlmath{2xh+h^{2}}

Step 5 — Divide by h

Divide every remaining numerator term by h.

2x+h\hlmath{2x+h}