Mathematics · Ch 12 — Limits and Derivatives

Derivatives

12.5

Derivatives

The Meaning of a Derivative

The idea of a derivative grows directly out of a practical question: how fast is something changing? If you know where a car is at different times, you can work out its speed. If you know the depth of water in a reservoir at several moments, you can predict when it will overflow. Rocket scientists need the velocity of a satellite at the instant it leaves the rocket, knowing the rocket's height over time. In every case, we want to know how one quantity changes with respect to another.

The derivative of a function at a point is the precise mathematical tool that answers this. It measures the instantaneous rate of change of the function at that point.

Definition of the Derivative at a Point

Suppose $f$ is a real-valued function and $a$ is a point in its domain. The derivative of $f$ at $a$ is defined as

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

provided this limit exists. The notation $f'(a)$ is read as "f-prime of a". This limit quantifies exactly how $f(x)$ is changing at the specific point $x = a$ .

Watch out

The denominator is $h$ , not $x$ . A common mistake is to write $f(a+h)-f(a-h)$ or to forget that the numerator must be the change in the function value over an interval of length $h$ .

Geometric Interpretation

Let $y = f(x)$ be a function. Take two points on its graph: $P = (a, f(a))$ and $Q = (a+h, f(a+h))$ , where $h$ is a small non-zero number. Draw the chord $PQ$ . In triangle $PQR$ (where $R$ is the point directly below $Q$ at the same height as $P$ ), the ratio

\frac{QR}{PR} = \frac{f(a+h) - f(a)}{h}

is exactly $\tan(\angle QPR)$ , which is the slope of the chord $PQ$ .

Now let $h$ approach $0$ . The point $Q$ slides along the curve towards $P$ , and the chord $PQ$ rotates until it becomes the tangent line to the curve at $P$ . The limit of the slope of the chord is therefore the slope of the tangent. Hence

f'(a) = \tan \psi

where $\psi$ is the angle the tangent line makes with the positive $x$ -axis. The derivative at a point is the slope of the tangent to the curve at that point.

Note

This geometric view is why derivatives are so powerful: they connect algebra (the limit of a ratio) with geometry (the slope of a tangent).

The Derivative as a Function

If the derivative exists at every point $x$ in the domain of $f$ , then it defines a new function called the derivative of $f$ . This is the formal definition.

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

This definition is also called the first principle of derivatives. The domain of $f'(x)$ is the set of all $x$ for which this limit exists. …

Definition 1Derivatives

The derivative of a real-valued function $f$ at a point $a$ in its domain is defined as:

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

provided this limit exists. The notation $f'(a)$ is read as "f-prime of a".

If the derivative exists at every point $x$ in the domain of $f$ , then it defines a new function, called the derivative of $f$ , given by:

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

wherever the limit exists. This is also called the first principle of derivative. The domain of $f'(x)$ is exactly the set of points where the above limit exists.

Note

The derivative is also denoted by $\frac{d}{dx}f(x)$ or, if $y = f(x)$ , by $\frac{dy}{dx}$ . At a specific point $x = a$ , you may see $\left.\frac{df}{dx}\right|_{x=a}$ or $\left.\frac{dy}{dx}\right|_{x=a}$ .

Intuition: The derivative measures the instantaneous rate of change of $f$ with respect to $x$ at a given point. Geometrically, $f'(a)$ is the slope of the tangent line to the curve $y = f(x)$ at the point $(a, f(a))$ .

Concrete example: For $f(x) = 3x$ , the derivative at $x = 2$ is: …

Definition 2Derivatives

The derivative of a real-valued function $f$ at a point $a$ in its domain is defined as:

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

provided this limit exists. The notation $f'(a)$ is read as "f-prime of a".

If the derivative exists at every point $x$ in the domain of $f$ , then it defines a new function, called the derivative of $f$ , given by:

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

wherever the limit exists. This is also called the first principle of derivative. The domain of $f'(x)$ is exactly the set of points where the above limit exists.

Note

Concrete example: For $f(x) = 3x$ , the derivative at $x = 2$ is: …

Figure 12.11Derivative as slope: chord PQ meets the x-axis at angle ψ, f′(a)=tan ψ

Fig. 12.11 — Derivative as slope: chord PQ meets the x-axis at angle ψ, f′(a)=tan ψ

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What Fig. 12.11 Shows

The figure plots a smooth, increasing curve $y = f(x)$ on standard $X$ - $Y$ axes. Two specific points are marked on this curve: $P = (a, f(a))$ and $Q = (a+h, f(a+h))$ , where $h$ is a small non-zero number. A dashed vertical guide drops from $P$ to the $x$ -axis at $x = a$ , and another from $Q$ to $x = a+h$ . Similarly, dashed horizontal guides run from $P$ and $Q$ to the $y$ -axis at $f(a)$ and $f(a+h)$ .

The secant line $PQ$ is drawn as a straight chord connecting the two points, and this line is extended until it meets the $x$ -axis. The angle that this extended secant makes with the positive $x$ -axis is labelled $\psi$ . A right triangle $PRQ$ is constructed: $P$ and $Q$ are the two points on the curve, and $R$ is the point directly below $Q$ and horizontally aligned with $P$ . The horizontal leg $PR$ has length $h$ (the change in $x$ ), and the vertical leg $RQ$ has length $f(a+h) - f(a)$ (the change in $y$ ).

The Physical Idea

The figure translates the abstract limit definition of a derivative into a concrete geometric picture. The ratio $\frac{f(a+h) - f(a)}{h}$ is exactly the slope of the secant line $PQ$ . In triangle $PRQ$ , this ratio equals $\tan(\angle QPR)$ , which is the same as $\tan \psi$ because $\psi$ is the angle the secant makes with the horizontal.

As $h$ shrinks toward zero, $Q$ slides along the curve toward $P$ . The secant $PQ$ rotates and approaches a limiting line — the tangent to the curve at $P$ . The slope of this tangent is the limit of the secant slopes, which is precisely $f'(a)$ . The figure shows that this limiting slope equals $\tan \psi$ , where $\psi$ is now the angle the tangent makes with the $x$ -axis.

Important

The derivative $f'(a)$ is the slope of the tangent line to $y = f(x)$ at $x = a$ . Geometrically, $f'(a) = \tan \psi$ , where $\psi$ is the angle the tangent makes with the positive $x$ -axis.

The Key Formula

The textbook develops the definition of the derivative from this figure:

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

Each symbol means:

$f'(a)$ — the derivative of $f$ at the point $x = a$
$h$ — a small increment in $x$ (the horizontal distance $PR$ in the triangle)
$f(a+h) - f(a)$ — the corresponding change in $y$ (the vertical leg $RQ$ )
$\frac{f(a+h) - f(a)}{h}$ — the slope of the secant $PQ$ , equal to $\tan \psi$ for the secant …

Start your 14-day free trial to unlock the full solution →

← Previous · 12.4Q32 — If f(x) = mx² + n, x < 0 nx + m, 0 ≤ x ≤ 1 nx³ + m, x > 1 . For what i…Next · 12.5 →Example 5 — Find the derivative at x = 2 of the function f(x) = 3x.