# Derivative & Antiderivative - Graphical Analysis

In summary: Thank you for catching that!In summary, the blue curve represents the derivative of the red curve, while the red curve represents the antiderivative. When the derivative is negative, the red curve is decreasing, while when the derivative is positive, the red curve is increasing.
The general idea I have in mind when it comes to analyzing a graph that has a derivative $$f$$ and its antiderivative $$F$$ ($$\mathrm{C}=0$$) is the following:

- When $$f$$ is positive, $$F$$ increases.
- When $$f$$ is negative, $$F$$ decreases.

However, I came across a couple of problems in my textbook that don't seem to work exactly that way. That's when either the derivative or the integral is an even function. Here is what I'm talking about:

$$\int \frac{x}{\sqrt{x^2 +1}} \: dx = \sqrt{x^2 +1} + \mathrm{C} \qquad (1)$$

$$\int \tan ^2 \theta \sec ^2 \theta \: d\theta = \frac{\tan ^3 \theta}{3} + \mathrm{C} \qquad (2)$$

The graphs are located at http://photos.yahoo.com/thiago_j

Note: the blue curves represent $$f$$, while the red ones represent $$F$$. The elements from Eq. (1) are depicted in "calculus-5-5---34" while those from Eq. (2) appear in "calculus-5-5---36".

Is this correct?

Do I need to modify the domain so that I only show the part of the plot that work as expected?

Thank you very much

What do you think is wrong with the curves for #1? f is the integrand represented by the blue curve and R is the antiderivative represented by the red curve, so the blue curve is the derivative of the red curve

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I m trying to understand your problem, but i can't see our point. The graphs of the functions seems to be ok. on equation (1), $$f$$ is negative from $$(-\infty,0)$$ and $$F$$ is decreasing on that interval. And also, $$f$$ is positive on $$(0,+\infty)$$ and $$F$$ is increasing. Same is true for the second one. I think nothing is wrong with the graphs, or maybe i just didn't get your problem.

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Thats exactly what's happening in this plot.. When the derivative is negative, the red graph is decreasing, when its positive, its decreasing..

whozum said:
Thats exactly what's happening in this plot.. When the derivative is negative, the red graph is decreasing, when its positive, its decreasing..

I think you meant to say "when it's postive, it's increasing"!

Well, I've just taken a second look at both graphs and it now makes sense. I was a little confused. Thanks for your input, guys.

HallsofIvy said:
I think you meant to say "when it's postive, it's increasing"!

Yup, I really should re-read my posts before submitting.

## 1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is commonly denoted as f'(x) or dy/dx.

## 2. How is a derivative graphically represented?

A derivative can be graphically represented as the slope of a tangent line to the function's curve at a specific point. It is the instantaneous rate of change of the function at that point.

## 3. What is an antiderivative?

An antiderivative is the inverse operation of a derivative. It is a function whose derivative is equal to the original function.

## 4. How is an antiderivative graphically represented?

An antiderivative can be graphically represented as the area under the curve of the original function on a given interval. It is the accumulation of all the small changes in the original function.

## 5. What is the relationship between a derivative and an antiderivative?

The derivative and antiderivative are closely related, as the derivative of an antiderivative is equal to the original function. In other words, the derivative "undoes" the antiderivative and vice versa.

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