Derivatives... Life makes no sense

Can some one please (in simple terms) put reasoning behind why:

A: the derivative of the area of a circle=the circumference $\frac{d}{dx} \pi r^2=2\pi*r$

B: the derivative of the volume of a sphere=the surface area $\frac{d}{dx} 4/3\pi r^3=4\pi*r^2$

Yes I know what derivatives are. But I'm guessing that the two above statements aren't just coincidences, there has to be a reason for this.

#Calculus #Differentiation #HelpMe! #Proofs #Easymoney

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Instead of looking at the circle's circumference as the derivative (lit. rate of change) of the area of a circle, you could view the area of the circle as the definite integral of the circumference as the infinitesimal radial element varies from 0 to r, or mathematically : $\displaystyle\int_{0}^{r} 2\pi r dr = \pi r^{2}$

This seems more intuitively sensible since if you multiply the circumference by infinitely smaller radial measures from the origin(0) to the very end of the circle(r), and add them all up, you will get the 'thickness' of the circle(area).

Now, we see why the rate of change of the area of the circle should be it's circumference (because the rate of change is the sum of the circumferences of infinitesimal thicknesses $dr$ )

Along the same lines, the rate of change of volume of a sphere is it's surface area because the change in volume can really be represented as the sum of the surface areas of infinitesimal thicknesses $dr$ )

Hope that helped.

Warning : This might be more confusing than helpful since I've never written/explained things like this in calculus before.

P.S: in A. it should be the circumference, and not the surface area (guess you repeated that) and in both A. and B., it should be $dr$ , not $dx$ .

B.S.Bharath Sai Guhan - 6 years, 8 months ago

Well, take a circle of radius r. And a circle of radius (r+dr)

What is the difference between this two circles? It's only a bit of boundary. I mean, you can get the larger circle just by adding a very thin hollow ring onto the smaller circle. But the area of the ring is 2pir dr , since it's thickness is negligible.

Consider, the circle,

It's just the same thing here. Much like an onion, when you increase the volume of the sphere by adding a infinitesimally thin peel with surface area equal to that of the smaller sphere, we are doing the same thing by adding a hollow layer of volume 4pir^2 dr

Agnishom Chattopadhyay - 6 years, 8 months ago

That was a question worth asking. I will volunteer to explain again if you do not get it.

Agnishom Chattopadhyay - 6 years, 8 months ago

yes i too

i am not able to understand how $\frac{d}{dx}$ tells the rate of change

here also in a) how is the rate of change of area of circle $2\pi r$ . can you tell me an practical example

here to

i was amazed when i saw the application part

the question was

which is greater $tanx or x$

solution

let

$f(x) = tanx - x$

$f ' (x) = sec^{2} x - 1 = tan^{2}x$

since tanx is an increasing function and $tan^{2}x$ will remain postive for negative values of x too

therefore tanx is greater than x

now how the rate of change of $\frac{perpendicular}{base}$ is $\frac{hypotenuse}{base}$

and too how the rate of change of x is 1

please explain

U Z - 6 years, 7 months ago

@U Z – The rate of change of x is not 1. The rate of change of x with respect to the rate of change of x itself is 1.

Who said tan(x) is greater than x? $\tan (7 \frac{\pi}{4} ) = -1$ and $-1 < 7 \frac{\pi}{4}$ That interpretation only holds good as long as tan(x) is increasing. (It is not strictly increasing all the time)

Okay, so what are derivatives? They point out the rate of change of a small amount of f(x) with respect to a small amount of change in x. How small? Infinitesimally small.

In fact it is easier to perceive it as the slope of the tangent of the function at a particular point.

Agnishom Chattopadhyay - 6 years, 7 months ago

@Agnishom Chattopadhyay – I have some more detailed explanations on this topic. I'll upload them here as a note as soon as I can. I believe reading through them will clarify your doubts. Please let me know if you want me to upload them @megh choksi

Agnishom Chattopadhyay - 6 years, 7 months ago

@Agnishom Chattopadhyay – Yes it would be very helpful

U Z - 6 years, 7 months ago

@U Z – Kindly read this note

It is a transcript from a person I taught Calculus.

Agnishom Chattopadhyay - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$