Question

What is the point of derivatives? Like what are we actually finding. Obviously it changes the entire function, but i don't really see why. I mean i can multiply, divide, or do random stuff to any function i want. What makes the derivative so special?

Answer 1

It helps me to understand and do math better when i know the logic behind it.

Answer 2

IE Real world use.

Answer 3

The instantaneous slope at a point.

Answer 4

They have lots of applications. Optimization of functions is one of them.

Answer 5

Differential Equations too. For example, by using Newton's law of cooling, we can predict the temperature of certain things. Radioactive decay and so on. They all are related to derivatives.

Answer 6

say you want to find the minimum value of \(7x+\frac{48}{x}\) over some interval, say \([1,20]\)
you do not have time to check every number and see which one gives you the smallest number
so you find out where the slope is zero and check those points

Answer 7

Hello there!

There are lots of reasons we'd want to take the derivative of
something. First of all, let's say you're riding in your shiny
new sports car and you have the best odometer in the world. It
will tell you to the nearest thousandth of a mile (or something
like that) how far you've gone. If you graphed what the odometer
tells you as a function of time, so that time is on the x-axis and
distance is on the y-axis, you could take the derivative of this
function and figure out your speed for every point in your journey.
So all the information about your speed and acceleration and
everything can be gotten from the odometer, as long as you know
how to take derivatives.

Here's a question my calculus teacher once asked me: in cars, there's
both an odometer and a speedometer. Essentially, the speedometer
takes the derivative of the odometer information (before it gets to
the odometer though; it's straight from the wheels). How does it do
that? It's been doing that since way before on-board computers
happened to cars. So essentially, they've found a purely mechanical
way to take derivatives. Neat stuff, worth researching.

The derivative is also quite an intuitive concept, I think. Let's
say you have a growth chart on your wall. If you're a human (which I
believe you are) you'll probably have a couple of periods when you
grew faster than at other times in your life. If the marks were made
at regular intervals, they'd be more spread out in certain periods
and more clustered together in others. So it's not hard to figure out
from this chart that you grew faster in those growth spurt times than
in the lull times. Well, how fast you grew is just the derivative
with respect to time of how tall you were. So the derivative will be
big sometimes, small sometimes, and once you hit 40 years old, it
will be negative (some people say).

So these are a couple of real-life examples. Other examples that
are based on integration (the inverse of differentiation) would
include finding the volume of some objects, finding the area of
some regions in a plane, and stuff like that. And trust me, if you
go on and do some more in math, taking the derivative of functions
will be SHEER BLISS compared with some of the more nasty stuff
(which is more rewarding. Stick with math!).

So that's how I feel about derivatives.

Answer 8

The derivative tells you the best local linear approximation of a function at a point.