# Derivatives in relation to curve sketching

• scorpa
In summary, the first question is to find the first derivative of x=x^2+1. I found it using the quotient rule and then solved it. The second question asks if f(x) = x^2(1-x). I found that it is concave upward and used the second derivative to find this. Everything is correct so far. I am on the right track for the first question, but I will redo the last part of the first question to be sure. Thanks for the help!
scorpa
Hi Again!

Right now I'm taking derivatives in relation to curve sketching, and I just wanted to make sure I am doing these right.

The first question is to determine using the first derivative where the graph of y = x is rising.
(x^2)+1

This is what I have done so far:

I tried to find the derivative using the quotient rule...

((x^2)+1)(d/dx)(x)-(x)(d/dx)((x^2)+1)
((x^2)+1)^2

((x^2)+1)(1)-(x)(2x)
((x^2)+1)^2

(-x^2)+1
((x^2)+1)^2

That is the value I found for the first derivative, although I am unsure whether I have done it right or not. Then to find where the curve was rising I said that since the derivative must be greater than 0, the value I found for the derivative must be greater than zero. Then I tried solving that and I ended up with -x^2 + 1 > x^4 + 2x^2 + 2 and although I realize that that is not quite finished yet, it just doesn't seem right to me. I am obviously going wrong somewhere, but where I do not know.

The second question asks you to determine where f(x) = x^2(1-x) is concave upward using the second derivative. This is what I did:

(x^2)(d/dx)(1-x) + (1-x)(d/dx)(x^2) =

(x^2)(-1) + (1-x)(2x) =

(-3x^2) + 2x This is the first derivative I found using the product rule and the steps shown above.

Then I found the second derivative to be -6x + 2

Have I found the derivatives correctly for these questions? It just seems like something is very wrong, mostly with the first question. Any help you guys can give me I would really appreciate. Thanks in advance.

a) The first derivative is correct.However,you solved this inequation

$$\frac{-x^{2}+1}{\left(x^{2}+1\right)^{2}} >0$$

incorrectly...

Do it again HINT:The denominator is always positive

For the second,everything is correct so far.How do you interpret the result...?

Daniel.

Thank you so much for the help, I really appreciate it!

Ok for the concave one ( second question) I think it is concave upward from (-infinity, (1/3))

I solved it like this:

-6x+2>0
-6x>-2
x < (1/3)

I'm redoing the last part of the first question right now, I post again in a minute.

Okay.It's okay,so far.

Daniel.

OK now for the first one,

I went back and did this:

(-x^2)+1 >0
((x^2)+1)^2

=

x^2 < 1

x<1
x>-1

therefore -1 < x < 1

OK now an I on the right track? Thanks again for the help!

It's perfect,u can plot it to get a graphical confirmation,but it's everything okay now.

Daniel.

Alright! Thank you so much!

## 1. What are derivatives?

Derivatives are mathematical tools used to measure the rate of change of a function. They can tell us how much a function is changing at a specific point or over a certain interval.

## 2. How are derivatives related to curve sketching?

Derivatives are essential in curve sketching because they help us determine critical points, where the slope of a function changes from positive to negative or vice versa. These critical points are important in determining the shape of a curve.

## 3. What is the process of finding derivatives?

The process of finding derivatives involves taking the limit of the difference quotient as the change in x approaches zero. This results in the derivative, which is the slope of the tangent line to the function at a specific point.

## 4. How can derivatives help us analyze a curve?

Derivatives can help us analyze a curve by giving us information about the slope of the curve at different points. They can also help us identify maximum and minimum points, inflection points, and concavity of the curve.

## 5. Are there any shortcuts for finding derivatives?

Yes, there are some rules and formulas that can be used as shortcuts for finding derivatives. These include the power rule, product rule, quotient rule, and chain rule. It is important to understand and practice these rules to make the process of finding derivatives faster and more efficient.

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