Derivatives: Definition, Interpretation, and Applications

Derivatives

Definition of the Derivative at a Point

The derivative of a function at a point measures the instantaneous rate of change of the function at that point. It is defined as the limit of the average rate of change (the slope of the secant line) as the interval shrinks to zero.

• Formal Definition: The derivative of f at x = a is given by:

$f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h}$

• This limit, if it exists, gives the slope of the tangent line to the graph of f at x = a.

• Alternative Definition:

$f'(a) = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}$

• Both definitions are equivalent and describe the same concept.

Geometric Interpretation

The derivative at a point corresponds to the slope of the tangent line to the curve at that point. The secant line between two points approaches the tangent line as the points get closer together.

• Secant Line Slope: The average rate of change between two points (x_1, y_1) and (x_2, y_2) is:

$m_{secant} = \frac{y_2 - y_1}{x_2 - x_1}$

• As x_2 approaches x_1, the secant line becomes the tangent line.

Terminology and Notation

• "Differentiate f at a": Find f'(a).

• "Differentiable at a": f'(a) exists.

• "Differentiability": The property of a function being differentiable at a point or on an interval.

• Common Notations for the Derivative at a Point:

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

Example: Derivative from the Definition

Example: Let $f(x) = \sqrt{2x}$. Find $f'(2)$ and the equations of:

• (a) The tangent line to $y = f(x)$ at $x = 2$

• (b) The normal line to $y = f(x)$ at $x = 2$

Solution:

First, compute the derivative at $x = 2$ using the definition:

$f'(2) = \lim_{h \rightarrow 0} \frac{\sqrt{2(2+h)} - \sqrt{4}}{h}$ $= \lim_{h \rightarrow 0} \frac{\sqrt{4+2h} - 2}{h}$ $= \lim_{h \rightarrow 0} \frac{\sqrt{4+2h} - 2}{h} \cdot \frac{\sqrt{4+2h} + 2}{\sqrt{4+2h} + 2}$ $= \lim_{h \rightarrow 0} \frac{(4+2h) - 4}{h(\sqrt{4+2h} + 2)}$ $= \lim_{h \rightarrow 0} \frac{2h}{h(\sqrt{4+2h} + 2)}$ $= \lim_{h \rightarrow 0} \frac{2}{\sqrt{4+2h} + 2}$ $= \frac{2}{2 + 2} = \frac{1}{2}$

• Tangent Line: The equation of the tangent line at $x = 2$ is:

$y - f(2) = f'(2)(x - 2)$ $y - 2 = \frac{1}{2}(x - 2)$ $y = \frac{1}{2}x + 1$

• Normal Line: The normal line is perpendicular to the tangent, so its slope is $-2$ (the negative reciprocal of $\frac{1}{2}$):

$y - 2 = -2(x - 2)$ $y = -2x + 6$

Derivative as a Function

The derivative can be defined as a function, giving the instantaneous rate of change at every point in the domain where the function is differentiable.

• Definition:

$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$ or $f'(x) = \lim_{z \rightarrow x} \frac{f(z) - f(x)}{z - x}$

• This function $f'(x)$ can be graphed and analyzed just like any other function.

Example: Derivative as a Function and Horizontal Tangents

Example: Let $f(x) = x^3 - x$. Find $f'(x)$ from the definition and determine where the tangent line is horizontal.

Solution:

$f'(x) = \lim_{h \rightarrow 0} \frac{(x+h)^3 - (x+h) - (x^3 - x)}{h}$ $= \lim_{h \rightarrow 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x - h - x^3 + x}{h}$ $= \lim_{h \rightarrow 0} \frac{3x^2h + 3xh^2 + h^3 - h}{h}$ $= \lim_{h \rightarrow 0} (3x^2 + 3xh + h^2 - 1)$ $= 3x^2 - 1$

• Horizontal Tangent: The tangent is horizontal where $f'(x) = 0$:

$3x^2 - 1 = 0$ $x^2 = \frac{1}{3}$ $x = \pm \sqrt{\frac{1}{3}}$

Differentiability and Non-Differentiable Points

A function is differentiable at a point if its derivative exists at that point. Some functions, such as those with sharp corners or cusps, are not differentiable at certain points.

• Example: Is $f(x) = |x|$ differentiable at $x = 0$?

Solution:

$f'(0) = \lim_{h \rightarrow 0} \frac{|0 + h| - |0|}{h} = \lim_{h \rightarrow 0} \frac{|h|}{h}$

• For $h > 0$, $\frac{|h|}{h} = 1$

• For $h < 0$, $\frac{|h|}{h} = -1$

• The left and right limits are not equal, so the limit does not exist.

Therefore, $f(x) = |x|$ is not differentiable at $x = 0$.

Summary Table: Differentiability

Function	Point	Differentiable?	Reason
$f(x) = x^2$	All $x$	Yes	Polynomial, smooth everywhere
$f(x) = |x|$	$x = 0$	No	Sharp corner at $x = 0$
$f(x) = \sqrt{x}$	$x = 0$	Yes (from right)	Defined and smooth for $x \geq 0$

Key Points

• The derivative at a point gives the slope of the tangent line at that point.

• The derivative as a function describes the instantaneous rate of change at every point in the domain.

• Not all functions are differentiable everywhere; points of non-differentiability include sharp corners, cusps, and discontinuities.

• Common notations for the derivative include $f'(x)$, $\frac{df}{dx}$, and $\frac{dy}{dx}$.