Intuition on intro to derivatives

Hello, I've just began to learn calculus and have recently come across derivatives.

It was all going well until it was said that finding the derivative of a graph will find the slope at any point. I understand the lim h --> 0 = f(x+h) - f(x) / h , it makes a lot of sense to me and I'm extremely happy with it but I just don't understand how the derivative of a graph can find the slope at any point as it's constantly changing.

Is there any bit of information I am missing? I'm self taught so there could well be something inuitive which I have missed.

Is the derivative the slope as f(x) changes with respect to x? If so then when I think of cases where f(x) really fluctuates to extremes I don't understand how f(x) changes with respect to x at the same rate.

I'm sure there is some information I completely misinterpreted. Is there a video which could provide me with the inuition or is there something I got wrong?

Thank you

(edited 7 years ago)

Reply 1

HapaxOromenon3

3

Original post by --wispr

Hello, I've just began to learn calculus and have recently come across derivatives.

It was all going well until it was said that finding the derivative of a graph will find the slope at any point. I understand the lim h --> 0 = f(x+h) - f(x) / h , it makes a lot of sense to me and I'm extremely happy with it but I just don't understand how the derivative of a graph can find the slope at any point as it's constantly changing.

Is there any bit of information I am missing? I'm self taught so there could well be something inuitive which I have missed.

Is the derivative the slope as f(x) changes with respect to x? If so then when I think of cases where f(x) really fluctuates to extremes I don't understand how f(x) changes with respect to x at the same rate.

I'm sure there is some information I completely misinterpreted. Is there a video which could provide me with the inuition or is there something I got wrong?

Thank you

The derivative is a function that gives the gradient of the curve at any specific point, in terms of its x-coordinate. For a complete development and explanation of derivatives, look at pages 1-50ish of https://www.math.wisc.edu/formMail/throttle.php?URL=/~keisler/keislercalc-12-20-15.pdf

Original post by --wispr

Hello, I've just began to learn calculus and have recently come across derivatives.

It was all going well until it was said that finding the derivative of a graph will find the slope at any point. I understand the lim h --> 0 = f(x+h) - f(x) / h , it makes a lot of sense to me and I'm extremely happy with it but I just don't understand how the derivative of a graph can find the slope at any point as it's constantly changing.

The limit is the formal definition for the derivative of a function - it is literally built upon the idea of finding the gradient between 2 points. It can be used to find the slope at a point

x=a

which means substituting an x-coordinate into the limit. The gradient along a curve will vary as the x-coordinate varies.

(edited 7 years ago)

Reply 3

GPiph

16

This sounds very interesting.

Reply 4

MartyO

7

Here's an animation on how the limit is used to find the slope of the tangent line to the curve (it is for the slope at one point though

\lim _{ x\rightarrow a }{ \frac { f(x)-f(x) }{ x-a } }

):
http://i.imgur.com/ffqJfd1.gif

In this other animation, the red function is the derivative of the green function, meaning it's image represents the gradient of the tangent line to the curve:
http://i.imgur.com/Iav3O27.gif

Intuition on intro to derivatives

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