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

In summary, the relation between the graph of a function and the graph of its derivative is that the derivative shows the rate of change of the function at a given point. By graphing both the function and its derivative on the same coordinate axes, we can determine the behavior of the function, such as whether it is increasing or decreasing, concave up or concave down, and where it has maxima or minima or inflection points. This can be done by simply looking at the sign of the first derivative without substituting any values. The equation of the derivative provides this information, and by looking at its graph, we can make conclusions about the original function.
  • #1
Juwane
87
0
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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  • #2
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.
 
  • #3
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)?
 
  • #4
Exactly what I said.
 
  • #5
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?
 
  • #6
Again, exactly what I said.
 

1. What is the relationship between the graph of a function and the graph of its derivative?

The graph of a function and its derivative are closely related because the derivative represents the rate of change of the function at any given point. This means that the slope of the tangent line to the function's graph at a specific point is equal to the value of the derivative at that point. In other words, the derivative gives us information about the steepness of the function's graph at different points.

2. How can the graph of a function's derivative be used to analyze the original function?

The graph of a function's derivative can be used to analyze the original function in several ways. For example, the derivative can tell us where the function is increasing or decreasing, where it has maximum or minimum values, and where it has points of inflection. This information can help us understand the behavior of the function and make predictions about its values.

3. Can the graph of a function's derivative have any gaps or discontinuities?

Yes, the graph of a function's derivative can have gaps or discontinuities. This can occur when the original function has a sharp change in slope or when it has points where it is not differentiable. In these cases, the derivative may have a jump or a vertical asymptote in its graph.

4. What is the relationship between the concavity of a function and the graph of its second derivative?

The concavity of a function is related to the graph of its second derivative. If the second derivative is positive, the function is concave up, meaning it has a U-shape. If the second derivative is negative, the function is concave down, meaning it has an upside-down U-shape. The points where the second derivative changes sign are called points of inflection on the graph of the original function.

5. Can the graph of a function and its derivative intersect at any point?

No, the graph of a function and its derivative cannot intersect at any point. This is because the derivative represents the slope of the tangent line to the function's graph at a specific point, so it will never have the same value as the function's graph at that point. However, they can have common points, such as points of inflection.

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