The Derivative

Derivative from the Limit Definition

Take lim_{h→0} of the simplified difference quotient to find f'(x)=2x for f(x)=x².

Example

Use the limit definition to derive the derivative exactly. This limit is the definition of the derivative; the shortcut rules in later lessons are theorems proved from it, which this book states rather than proves.

highlighted = computed this step

Step 1 — Set up

Start with the limit definition of the derivative.

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x)=\lim_{h\to 0 }\frac{f(x+h)-f(x)}{h}

Step 2 — Expand f of x plus h

Substitute f of x plus h and f of x.

lim⁡h→0(x+h)2−x2h\lim_{h\to 0 } \hlmath{\frac{(x+h)^{2}-x^{2}}{h}}

Step 3 — Expand numerator

Expand the numerator to get 2xh plus h squared.

lim⁡h→02xh+h2h\lim_{h\to 0 } \hlmath{\frac{2xh+h^{2}}{h}}

Step 4 — Cancel h

Cancel the common h before taking the limit.

lim⁡h→02x+h\lim_{h\to 0 } \hlmath{2x+h}

Step 5 — Take h to zero

Let h go to 0 to get 2x.

f′(x)=2xf'(x)= \hlmath{2x}
derivative-definition The derivative f'(x) is defined as lim_{h→0} [f(x+h)-f(x)]/h. For f(x)=x² the difference quotient simplifies to 2x+h, and as h→0 the limit is 2x — the slope of the tangent at any point x.