CalculusConcept reference

Derivative definition

The derivative at a point is the finite two-sided limit of its difference quotient.

On this page 7 sections

01At input a, the difference quotient Q(h) compares nearby outputs#

At input $a$ , the difference quotient $Q(h)$ compares nearby outputs. Keep $h\ne0$ while forming it:

\frac{f(a+h)-f(a)}h

The derivative is the finite two-sided limit $f\prime(a)=\lim_{h\to0}Q(h)$ .

For $f(x)=x^2$ at $a=2$ , expand $(2+h)^2=4+4h+h^2$ . Subtract $f(2)=4$ to get the output change:

4h+h^2

For $f(x)=x^2$ at $2$ , divide its increment $4h+h^2$ by $h\ne0$ . Cancel the shared factor, then take the limit.

Q(h)=4+h\ \longrightarrow\ 4

Thus $f\prime(2)=4$ . Substituting zero before canceling would leave $0/0$ .

Check your reasoning

$f(x)=x^{2}+2x+1$ , $a=1$ . Find the quotient $Q(h)$ and slope.

Show answer and explanation

$Q=4+h$ ; slope $4$ .

For $h\ne0$ , $Q=4+h\to 4$ .

The exact quotient $Q(h)=4+h$ approaches $4$ from either side. Samples illustrate this; the simplified formula establishes the limit.

Check your reasoning

A response changes by $\Delta y=5h-2h^{2}$ . Find $\lim_{h\to0}\Delta y/h$ .

Show answer and explanation

Divide by $h$ : $5-2h\to5$ .

Check your reasoning

$f(x)=x^{2}-x+4$ , $a=3$ . Claim: set $h=0$ first. Repair the quotient and derivative.

Show answer and explanation

$Q=5+h$ ; slope $5$ .

For $h\ne0$ , $Q=5+h\to 5$ .

Key takeaway