KAHIBARO
Table of Contents
Change and Slope
In physics, many quantities change. Position changes with time, temperature changes with location, and voltage changes from point to point in a circuit. A derivative is the mathematical tool that tells us how fast one quantity changes compared with another.
The simplest way to understand a derivative is to think about the slope of a graph. If a graph shows a quantity $y$ as a function of $x$, then the derivative tells us the slope at a single point. This slope describes how sensitive $y$ is to small changes in $x$.
If the slope is positive, $y$ increases as $x$ increases. If the slope is negative, $y$ decreases as $x$ increases. If the slope is zero, the graph is flat at that point.
Average Change and Instantaneous Change
Suppose a function changes from $f(x)$ to $f(x+\Delta x)$. The average rate of change over that interval is
$$
\frac{\Delta f}{\Delta x} = \frac{f(x+\Delta x)-f(x)}{\Delta x}.
$$
This gives the slope of a secant line, which connects two points on the curve. But in physics we often want the rate of change at one exact instant or one exact position. For that, we let the interval become very small.
The derivative of $f(x)$ is defined by the limit
$$
f'(x) = \lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}.
$$
This is the slope of the tangent line to the curve at the point $x$.
The derivative of a function $f(x)$ is
$$
f'(x) = \lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}.
$$
It gives the instantaneous rate of change of $f$ with respect to $x$, and the slope of the tangent line to the graph.
Notation for Derivatives
There are several common ways to write a derivative. If $y = f(x)$, then the derivative can be written as
$$
f'(x), \qquad y', \qquad \frac{dy}{dx}.
$$
These all mean the same thing, though they are used in slightly different contexts. The notation $\frac{dy}{dx}$ is especially common in physics because it clearly shows which variable is changing with respect to which other variable.
For higher derivatives, we write
$$
\frac{d^2 y}{dx^2}, \qquad \frac{d^3 y}{dx^3},
$$
and so on.
Geometric Meaning
A derivative is closely tied to the geometry of a graph. If the graph is steeply rising, the derivative is large and positive. If it is steeply falling, the derivative is large and negative. If it has a horizontal tangent, the derivative is zero there.
A derivative can also tell us about special points. For example, where a function reaches a local maximum or minimum, its derivative is often zero.
Basic Rules of Differentiation
Instead of using the limit definition every time, we usually use standard derivative rules.
If $c$ is a constant, then
$$
\frac{d}{dx}(c) = 0.
$$
If $x$ is the variable itself, then
$$
\frac{d}{dx}(x) = 1.
$$
If a function is multiplied by a constant $c$, then
$$
\frac{d}{dx}\bigl(c f(x)\bigr) = c f'(x).
$$
If two functions are added or subtracted, then
$$
\frac{d}{dx}\bigl(f(x) \pm g(x)\bigr) = f'(x) \pm g'(x).
$$
Important linearity rules:
$$
\frac{d}{dx}(c) = 0, \qquad \frac{d}{dx}\bigl(c f(x)\bigr) = c f'(x),
$$
$$
\frac{d}{dx}\bigl(f(x) \pm g(x)\bigr) = f'(x) \pm g'(x).
$$
Power Rule
One of the most useful rules in physics is the power rule. For any power of $x$,
$$
\frac{d}{dx}(x^n) = n x^{n-1}.
$$
This works for positive integers, negative integers, and many other real values of $n$ where the function is defined.
Examples are
$$
\frac{d}{dx}(x^2) = 2x,
$$
$$
\frac{d}{dx}(x^3) = 3x^2,
$$
$$
\frac{d}{dx}\left(\frac{1}{x}\right) = \frac{d}{dx}(x^{-1}) = -x^{-2} = -\frac{1}{x^2}.
$$
Power rule:
$$
\frac{d}{dx}(x^n) = n x^{n-1}.
$$
Product Rule
Sometimes a quantity is the product of two functions. Then we use the product rule:
$$
\frac{d}{dx}\bigl(f(x)g(x)\bigr) = f'(x)g(x) + f(x)g'(x).
$$
For example, if
$$
y = x^2 \sin x,
$$
then
$$
\frac{dy}{dx} = 2x \sin x + x^2 \cos x.
$$
A common mistake is to think the derivative of a product is just the product of derivatives. That is not true in general.
Product rule:
$$
\frac{d}{dx}\bigl(fg\bigr) = f'g + fg'.
$$
In general,
$$
\frac{d}{dx}(fg) \ne f'g'.
$$
Quotient Rule
If one function is divided by another, we use the quotient rule:
$$
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2},
\qquad g(x) \ne 0.
$$
For example, if
$$
y = \frac{x^2+1}{x},
$$
then
$$
\frac{dy}{dx}
= \frac{(2x)(x) - (x^2+1)(1)}{x^2}
= \frac{2x^2 - x^2 - 1}{x^2}
= \frac{x^2-1}{x^2}.
$$
Chain Rule
Very often in physics, one variable depends on another, which depends on another. In that case we use the chain rule.
If
$$
y = f(u), \qquad u = g(x),
$$
then
$$
\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}.
$$
In function notation, this is
$$
\frac{d}{dx}f(g(x)) = f'(g(x))g'(x).
$$
For example,
$$
\frac{d}{dx}(3x+1)^5 = 5(3x+1)^4 \cdot 3 = 15(3x+1)^4.
$$
The chain rule is one of the most important tools in all of physics and calculus.
Chain rule:
If $y = f(g(x))$, then
$$
\frac{dy}{dx} = f'(g(x)) \, g'(x).
$$
Equivalently,
$$
\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}.
$$
Common Derivatives
Here are some derivatives that appear often in physics.
| Function | Derivative |
|---|---|
| $c$ | $0$ |
| $x$ | $1$ |
| $x^n$ | $n x^{n-1}$ |
| $e^x$ | $e^x$ |
| $\ln x$ | $\frac{1}{x}$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
These should gradually become familiar through use.
Worked Examples
Consider
$$
f(x) = 4x^3 - 2x^2 + 7x - 5.
$$
Differentiate term by term:
$$
f'(x) = 12x^2 - 4x + 7.
$$
Now consider
$$
y = (x^2+1)(x^3-2).
$$
Using the product rule,
$$
\frac{dy}{dx} = (2x)(x^3-2) + (x^2+1)(3x^2).
$$
Now consider
$$
y = \frac{\sin x}{x}.
$$
Using the quotient rule,
$$
\frac{dy}{dx} = \frac{(\cos x)(x) - (\sin x)(1)}{x^2}
= \frac{x\cos x - \sin x}{x^2}.
$$
Finally, consider
$$
y = \sqrt{1+x^2} = (1+x^2)^{1/2}.
$$
Using the chain rule,
$$
\frac{dy}{dx}
= \frac{1}{2}(1+x^2)^{-1/2}(2x)
= \frac{x}{\sqrt{1+x^2}}.
$$
Derivatives and Units
In physics, derivatives often carry units. If position $x$ is measured in meters and time $t$ in seconds, then $\frac{dx}{dt}$ has units of meters per second. If temperature $T$ changes with distance $x$, then $\frac{dT}{dx}$ may have units of degrees per meter.
This is important because derivatives are not just abstract slopes. They describe real physical rates of change.
When a Derivative Does Not Exist
A derivative does not always exist at every point. This can happen if the graph has a sharp corner, a vertical tangent, or a break.
For example, the function $f(x) = |x|$ has a sharp corner at $x=0$. Its slope from the left is $-1$, and its slope from the right is $+1$, so there is no single derivative there.
Why Derivatives Matter
Derivatives are central to physics because physical laws often relate quantities to their rates of change. Motion, growth, decay, oscillation, and fields are all described using derivatives.
You can think of a derivative as a local description. It tells you what a function is doing right here, at this point, or at this instant. That makes it one of the most powerful ideas in mathematical physics.
A derivative measures local change.
It tells how one quantity changes with respect to another at a specific point:
$$
\text{derivative} = \text{instantaneous rate of change} = \text{slope of tangent line}.
$$