Derivative Definitions

For $f(x)=x^2$, moving $x$ from $3$ to $4$ changes the output from $9$ to $16$, so the average rate of change is $(16-9)/(4-3)=7$. Moving instead from $3$ to $3.1$ changes the output from $9$ to $9.61$, giving an average rate of $(9.61-9)/0.1=6.1$.

Write the nonzero input change as $h$, keeping the starting point at $a=3$. Dividing the output change by $h$ gives:

$$\frac{f(3+h)-f(3)}h =\frac{(3+h)^2-9}h=6+h\quad(h\ne0).$$

This ratio is the difference quotient, the average rate of change between the two points.

Input change $h$	Difference quotient $6+h$
$1$	$7$
$0.1$	$6.1$
$0.01$	$6.01$
$-0.1$	$5.9$

The slopes approach $6$ as $h$ approaches zero from either side. At $h=0$, the quotient is undefined; the derivative comes from the limit, not from substituting zero into the quotient.

In the square-function view below, $P=(3,9)$ stays fixed while $Q=(3+h,(3+h)^2)$ moves with $h$; for $h\ne0$, the line through them is a secant whose slope is the difference quotient.

Functionf(x) = x² · a = 3f(x) = |x| · a = 0

CurveSecant PQTangent at P

Move Q with h

h = 1.00

P = (3.00, 9.00) · Q = (4.00, 16.00)Difference quotient: 7.00The secant slope is 6 + h. As Q approaches P from either side, the slope approaches 6.

Formal Definition

For a function $y=f(x)$, the derivative is the instantaneous rate of change defined by

$$f'(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h},$$

provided the limit is a finite real number. The ordinary two-sided derivative requires $f$ to be defined on an open interval around the point and the quotients from both sides to approach the same value. At a domain endpoint, specify a one-sided derivative. The limit definition and examples develop this distinction between a function value and a derivative.

For the square function, $f'(3)=6$. More generally, at any real $a$,

$$\frac{f(a+h)-f(a)}h =\frac{a^2+2ah+h^2-a^2}h=2a+h\quad(h\ne0),$$

so $f'(a)=2a$. Cancellation is valid for nonzero $h$; no division by zero is performed.

From a difference quotient to a local approximation

Slope of the Tangent Line

The difference quotient is the slope of a secant through $(a,f(a))$ and $(a+h,f(a+h))$. Its finite limit $m=f'(a)$ is the slope of the tangent at $(a,f(a))$.

Equation of the Tangent Line

With this slope and the point $(a,f(a))$, the tangent line is

$$y = f(a) + f'(a)(x - a).$$

At $a=3$ for $x^2$, this is $y=9+6(x-3)$. Taking $h=0.01$, the line predicts $9+6(0.01)=9.06$, while the exact value is $(3.01)^2=9.0601$. The error is $h^2=0.0001$.

More generally, differentiability is equivalent to

$$f(a+h)=f(a)+f'(a)h+r(h),\qquad \lim_{h\to0}\frac{r(h)}h=0.$$

The remainder is written $r(h)=o(h)$: its magnitude becomes negligible relative to $|h|$. Subtracting $f'(a)$ from the defining difference quotient gives this condition. For the square function at $3$, $r(h)=h^2$, but differentiability alone does not guarantee a quadratic error bound or accuracy far from $a$.

When a local slope does not exist

The same remainder identity implies $f(a+h)\to f(a)$, so differentiability implies continuity. The converse fails: $f(x)=|x|$ is continuous at $0$, but its difference quotient $|h|/h$ is $-1$ for $h<0$ and $1$ for $h>0$. The one-sided slopes disagree, so there is no two-sided derivative.

Return to the figure above and switch to $|x|$ at $0$. Move $h$ toward zero from both sides: unlike $x^2$ at $3$, the secant slopes do not approach the same value. At $h=0$, the difference quotient remains undefined for both functions.

Also, $f(x)=\sqrt[3]{x}$ has quotient $h^{-2/3}\to+\infty$ at $0$; a vertical tangent is not a finite derivative.

Interpretation of the Derivative

Rate of Change

The tangent slope $f'(a)$ gives the instantaneous output change per unit input change near $a$. The local approximation $f(a+h)\approx f(a)+f'(a)h$ expresses that rate as a prediction.

Physical Interpretation in Motion

If $f(t)$ is an object's position at time $t$, then $f'(a)$ is its velocity at $t=a$. Position in metres and time in seconds give a derivative in metres per second. For several inputs, the analogous local linear term uses a gradient and a displacement vector.

Equivalent Notations for Derivatives

For $y=f(x)$, these notations all denote the derivative with respect to $x$:

• $f'(x)$
• $y'$
• $\frac{dy}{dx}$
• $\frac{d}{dx} f(x)$
• $Df(x)$

Notation for Derivatives at a Specific Point

At $x=a$, write

$$f'(a) = \left. \frac{dy}{dx} \right|_{x=a} = \left. \frac{d}{dx} f(x) \right|_{x=a} = Df(a).$$

References

• tutorial.math.lamar.edu/pdf/calculus_cheat_sheet_derivatives.pdf
• personal.math.ubc.ca/~CLP/CLP1/clp_1_dc_text.pdf
• mitres_18_001_f17_full_book.pdf

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