Question

Explain derivatives graphically.

For example:

Say there is a function f(x) = x^2 and its derivative 2x.

So each point on the slope of the derivative is supposed to represent the slope of the line tangent at a certain point on the original function.

Say I choose an x-value on the derivative 1, so the point on the line would be (1,2).

Where on the original function would this be the represented slope of? As I understand it, the x-value 1 corresponds with a slope of 2, the x-value 2 corresponds with a slope of 4, etc. but how do I find the point on the original function where these

Answer 1

So each point on the slope of the derivative is supposed to represent the slope of the line tangent at a certain point on the original function.

Say I choose an x-value on the derivative 1, so the point on the line would be (1,2).

Where on the original function would this be the represented slope of? As I understand it, the x-value 1 corresponds with a slope of 2, the x-value 2 corresponds with a slope of 4, etc. but how do I find the point on the original function where these are the slopes of?

Answer 2

the original point is what you get when you replace \(x\) by \(1\) so in this case you could say
"the line tangent to the curve \(y=x^2\) at the point \((1,1)\) is 2

Answer 3

that is to say the point on the curve of \(y=f(x)\) at \(x=a\) is \((a,f(a))\) and the slope of the tangent line to the curve at \((a,f(a))\) is \(f'(a)\)

Answer 4

Very good, thank you satellite73