Derivatives: Measuring Change

Zoom into a curve until it looks straight. Its slope is the derivative.

16 minUpdated 21 October 2026rate of changelimittangentdifferentiation rules

Lesson at a glanceShow the key points

dy/dx = lim (Δx → 0) Δy/Δx, the slope of the tangent to the graph of y against x.
dy/dx > 0 where y increases with x, < 0 where it decreases and = 0 where the tangent is flat.
For small changes, Δy ≈ (dy/dx) Δx.
Common results: d(xⁿ)/dx = nxⁿ⁻¹, d(sin x)/dx = cos x, d(cos x)/dx = −sin x, d(eˣ)/dx = eˣ, d(ln x)/dx = 1/x.
Rules: d(cy)/dx = c dy/dx; d(u + v)/dx = du/dx + dv/dx; d(uv)/dx = u dv/dx + v du/dx.

Physics is full of “how fast” questions: how fast position changes, how fast velocity changes, how fast temperature drops. The derivative is the precise answer.

Rate of change, roughly

Take two quantities $y$ and $x$ linked so that each $x$ gives exactly one $y$ , and plot the graph. Nudge $x$ by a small amount $\Delta x$ ; $y$ responds with a change $\Delta y$ . A natural first guess for “the rate at which $y$ changes with $x$ ” is

\frac{\Delta y}{\Delta x},

which is the slope of the straight chord (the secant) joining A $(x, y)$ to B $(x + \Delta x,\ y + \Delta y)$ .

The trouble is that a curve’s steepness changes along the way. Between A and B it might start gentle and end steep, and the chord averages all of that into one number. To pin down the rate at A itself we need B closer. And closer.

Interactive

From a secant to a tangent

A is fixed; B sits Δx further along. Shrink Δx and watch the chord AB swing into the tangent line, while Δy/Δx settles on a single number.

\Delta x

2.500

\Delta y

-2.450

\Delta y/\Delta x

-0.9800

dy/dx \text{ at } A

0.4156

Curve

Gap

\Delta x

2.500

As $\Delta x$ shrinks the chord pivots about A and settles onto the tangent line, and the ratio $\Delta y/\Delta x$ settles on a definite value even though both $\Delta y$ and $\Delta x$ are heading to zero. That limiting value is the derivative.

Derivative

\frac{dy}{dx} = \lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}

the slope of the tangent at that point

For a small but finite change you can turn this around and estimate: $\Delta y \approx \dfrac{dy}{dx}\,\Delta x$ . This approximation is how physicists reason about small changes all the time.

Interactive

Walk along the curve, draw its slope

Drag the point along the top curve. Its tangent slope is plotted below, so your path traces out the derivative. Green means rising, coral means falling, gold means flat.

x

1.00

y

4.48

dy/dx

2.16

Curve

Computing a derivative by hand

Graphs are great for intuition, but we usually know $y$ as a formula. Take the area of a square of side $L$ : $A = L^2$ . Grow the side to $L + \Delta L$ :

A + \Delta A = (L + \Delta L)^2 = L^2 + 2L\,\Delta L + (\Delta L)^2

\Rightarrow\ \frac{\Delta A}{\Delta L} = 2L + \Delta L \ \xrightarrow{\ \Delta L \to 0\ }\ 2L.

Interactive

How fast does a square’s area grow?

Stretch each side from L to L + ΔL. The new area is two thin strips (L·ΔL each) plus a tiny corner (ΔL²). As ΔL shrinks, the corner becomes negligible.

two strips 2L·ΔL4.80

corner ΔL²0.64

\Delta A/\Delta L = 2L + \Delta L

6.80

dA/dL = 2L

6.00

Side

L

3.0

Increase

\Delta L

0.80

So $\dfrac{d}{dL}(L^2) = 2L$ . The picture says why: the area grows mostly through two strips of length $L$ , and the little corner is a second-order crumb that vanishes in the limit.

A small library of derivatives

You will not do the limit by hand every time. Here are the results for the functions that show up most in physics. $dy/dx$ is also called the derivative or differential coefficient of $y$ with respect to $x$ .

y	dy/dx	y	dy/dx
$x^n$	$n x^{n-1}$	$\sec x$	$\sec x\tan x$
$\sin x$	$\cos x$	$\csc x$	$-\csc x\cot x$
$\cos x$	$-\sin x$	$\ln x$	$1/x$
$\tan x$	$\sec^2 x$	$e^x$	$e^x$
$\cot x$	$-\csc^2 x$

Three rules let you combine them:

\text{(a)}\ \frac{d}{dx}(c\,y) = c\frac{dy}{dx} \qquad \text{(b)}\ \frac{d}{dx}(u+v) = \frac{du}{dx}+\frac{dv}{dx} \qquad \text{(c)}\ \frac{d}{dx}(uv) = u\frac{dv}{dx}+v\frac{du}{dx}

Worked example

Product rule in action

Find $\dfrac{dy}{dx}$ for $y = e^x\cos x$ .

Worked example

Speed of a growing circle

A ripple spreads so that its radius is $r = (0.5\text{ m/s})\,t$ . How fast is the enclosed area growing at $t = 4\text{ s}$ ?

What to remember

1 $dy/dx = \lim_{\Delta x \to 0} \Delta y/\Delta x$ , which is the slope of the tangent.
2Positive derivative: rising. Negative: falling. Zero: momentarily flat.
3For small changes, $\Delta y \approx (dy/dx)\,\Delta x$ .
4Learn the table and the three rules (constant multiple, sum, product); they cover almost everything.

Check yourself

What is d/dx of 3x⁴ − 2x + 7?

On a graph of y against x, the tangent at some point is horizontal. What is dy/dx there?

If y = x², estimate Δy when x goes from 5 to 5.01.

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchThe essence of calculus3Blue1Brown WatchThe paradox of the derivative3Blue1BrownWhat does ‘instantaneous rate’ even mean?WatchDerivative formulas through geometry3Blue1BrownThe growing-square picture of d(x²)WatchDerivativesCrash Course Physics #2 PlayCalculus GrapherPhET · CU Boulder ReadDerivatives of common functionsHyperPhysics ReadThe paradox of the derivative (text lesson)3Blue1Brown

PreviousThe Cross Product Next Peaks & Valleys

Derivatives: Measuring Change

Zoom into a curve until it looks straight. Its slope is the derivative.

16 minUpdated 21 October 2026rate of changelimittangentdifferentiation rules

Lesson at a glanceShow the key points

dy/dx = lim (Δx → 0) Δy/Δx, the slope of the tangent to the graph of y against x.
dy/dx > 0 where y increases with x, < 0 where it decreases and = 0 where the tangent is flat.
For small changes, Δy ≈ (dy/dx) Δx.
Common results: d(xⁿ)/dx = nxⁿ⁻¹, d(sin x)/dx = cos x, d(cos x)/dx = −sin x, d(eˣ)/dx = eˣ, d(ln x)/dx = 1/x.
Rules: d(cy)/dx = c dy/dx; d(u + v)/dx = du/dx + dv/dx; d(uv)/dx = u dv/dx + v du/dx.

Physics is full of “how fast” questions: how fast position changes, how fast velocity changes, how fast temperature drops. The derivative is the precise answer.

Rate of change, roughly

\frac{\Delta y}{\Delta x},

which is the slope of the straight chord (the secant) joining A $(x, y)$ to B $(x + \Delta x,\ y + \Delta y)$ .

Interactive

From a secant to a tangent

A is fixed; B sits Δx further along. Shrink Δx and watch the chord AB swing into the tangent line, while Δy/Δx settles on a single number.

\Delta x

2.500

\Delta y

-2.450

\Delta y/\Delta x

-0.9800

dy/dx \text{ at } A

0.4156

Curve

Gap

\Delta x

2.500

Derivative

\frac{dy}{dx} = \lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}

the slope of the tangent at that point

For a small but finite change you can turn this around and estimate: $\Delta y \approx \dfrac{dy}{dx}\,\Delta x$ . This approximation is how physicists reason about small changes all the time.

Interactive

Walk along the curve, draw its slope

Drag the point along the top curve. Its tangent slope is plotted below, so your path traces out the derivative. Green means rising, coral means falling, gold means flat.

x

1.00

y

4.48

dy/dx

2.16

Curve

Computing a derivative by hand

Graphs are great for intuition, but we usually know $y$ as a formula. Take the area of a square of side $L$ : $A = L^2$ . Grow the side to $L + \Delta L$ :

A + \Delta A = (L + \Delta L)^2 = L^2 + 2L\,\Delta L + (\Delta L)^2

\Rightarrow\ \frac{\Delta A}{\Delta L} = 2L + \Delta L \ \xrightarrow{\ \Delta L \to 0\ }\ 2L.

Interactive

How fast does a square’s area grow?

Stretch each side from L to L + ΔL. The new area is two thin strips (L·ΔL each) plus a tiny corner (ΔL²). As ΔL shrinks, the corner becomes negligible.

two strips 2L·ΔL4.80

corner ΔL²0.64

\Delta A/\Delta L = 2L + \Delta L

6.80

dA/dL = 2L

6.00

Side

L

3.0

Increase

\Delta L

0.80

So $\dfrac{d}{dL}(L^2) = 2L$ . The picture says why: the area grows mostly through two strips of length $L$ , and the little corner is a second-order crumb that vanishes in the limit.

A small library of derivatives

y	dy/dx	y	dy/dx
$x^n$	$n x^{n-1}$	$\sec x$	$\sec x\tan x$
$\sin x$	$\cos x$	$\csc x$	$-\csc x\cot x$
$\cos x$	$-\sin x$	$\ln x$	$1/x$
$\tan x$	$\sec^2 x$	$e^x$	$e^x$
$\cot x$	$-\csc^2 x$

Three rules let you combine them:

\text{(a)}\ \frac{d}{dx}(c\,y) = c\frac{dy}{dx} \qquad \text{(b)}\ \frac{d}{dx}(u+v) = \frac{du}{dx}+\frac{dv}{dx} \qquad \text{(c)}\ \frac{d}{dx}(uv) = u\frac{dv}{dx}+v\frac{du}{dx}

Worked example

Product rule in action

Find $\dfrac{dy}{dx}$ for $y = e^x\cos x$ .

Worked example

Speed of a growing circle

A ripple spreads so that its radius is $r = (0.5\text{ m/s})\,t$ . How fast is the enclosed area growing at $t = 4\text{ s}$ ?

What to remember

1 $dy/dx = \lim_{\Delta x \to 0} \Delta y/\Delta x$ , which is the slope of the tangent.
2Positive derivative: rising. Negative: falling. Zero: momentarily flat.
3For small changes, $\Delta y \approx (dy/dx)\,\Delta x$ .
4Learn the table and the three rules (constant multiple, sum, product); they cover almost everything.

Check yourself

What is d/dx of 3x⁴ − 2x + 7?

On a graph of y against x, the tangent at some point is horizontal. What is dy/dx there?

If y = x², estimate Δy when x goes from 5 to 5.01.

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousThe Cross Product Next Peaks & Valleys

Rate of change, roughly#

Computing a derivative by hand#

A small library of derivatives#

Check yourself

Keep exploring

Rate of change, roughly#

Computing a derivative by hand#

A small library of derivatives#

Check yourself

Keep exploring

Rate of change, roughly

Computing a derivative by hand

A small library of derivatives

Rate of change, roughly

Computing a derivative by hand

A small library of derivatives