Relation of the graph of a function with the graph of it's derivative

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The discussion explores the relationship between the graph of a function, specifically x^2, and its derivative, 2x. It emphasizes that by analyzing the graph of the derivative, one can infer critical information about the original function, such as where it is increasing or decreasing, and identify points of maxima, minima, and inflection points. The conversation highlights that the slope of the tangent line at any point on the curve x^2 can be determined from the graph of its derivative, 2x, without needing to substitute specific values. It questions the utility of graphing the derivative if no insights can be gained about the original function from it. Ultimately, the relationship between the two graphs provides valuable insights into the behavior of the function.
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What is the relation between the graph of a function and the graph of it's derivative?

Suppose that the function is x^2. It's derivative is 2x. If we graph both x^2 and 2x on the same coordinate axes, can we conclude anything about x^2 by looking at the graph of 2x, or vice versa (i.e. can we conclude anything about 2x by looking at the graph of x^2)?

For example, can we tell what will be the slope of the tangent line at a curve at x^2 by looking at the graph of it's derivative 2x?

If there is no relation between the two, then what is the use of graphing the derivative of a function?
 
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You ought to be able to tell where it it is increasing, decreasing, concave up, concave down, where it has maxima or minima or inflection points just by looking at the sign of the first derivative and where it's increasing or decreasing.
 
rochfor1 said:
You ought to be able to tell where it it is increasing, decreasing, concave up, concave down, where it has maxima or minima or inflection points just by looking at the sign of the first derivative and where it's increasing or decreasing.

I'm not talking about when you find the slope of the tangent at some point by substituting the point into the the equation of the derivative. I'm talking about the equation of the derivative itself (such as y=2x for y=x^2).

I mean: what can we find out about x^2 just from the equation 2x (or it's graph) (without substituting any values)?
 
Exactly what I said.
 
Oops.. I think I'm asking the question the wrong way. My question is that by looking at the plot of 2x (which is a straight line), what can we say about the x^2 (which is a curve), without knowing that the straight line's equation is 2x and the curve's equation is x^2?
 
Again, exactly what I said.
 

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