Help with using the first derivative

• meeklobraca
In summary, to determine where the graph of y = x/(x^2+1) is rising, we can use the first derivative, which is (1-x^2) / (x^2+1)^2. We need to determine where the numerator is positive and where it is negative. By factoring 1 - x^2, we can see that it is positive for 0 < x < 1 and negative for x < 0 or x > 1. Therefore, the graph of y = x/(x^2+1) is rising for x between 0 and 1.
meeklobraca

Homework Statement

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

The Attempt at a Solution

Ive figured the derivative to be (1-x^2) / (x^2+1)^2 and I know that the derivative > 0 will tell me where the graph is rising. I am just not sure how to figure that out. Do I need to simplify my derivative a bit more to make it work?

The denominator will always be positive, so all you need to do is determine where the numerator is positive, and where negative. Factor 1 - x^2 and see where it is zero, and where positive, and where negative.

"Factor 1 - x^2 and see where it is zero, and where positive, and where negative."

Im not sure I know what you mean here. Factored form it is (x-1)(x+1)

Its 0 when x = +-1 , positive for 0<x<1 negative for (-infin,0) (1,infin)?

i think you may have missed the effect of one of the factors...

you could say (1-x^2) is +ve:
when
1-x^2 >0
implying
x^2 < 1

meeklobraca said:
"Factor 1 - x^2 and see where it is zero, and where positive, and where negative."

Im not sure I know what you mean here. Factored form it is (x-1)(x+1)

Its 0 when x = +-1 , positive for 0<x<1 negative for (-infin,0) (1,infin)?
"Factored form" is NOT (x- 1)(x+ 1), it is (1- x)(1+ x).

And the graph of y= (1-x)(1+x)= 1- x2, is a parabola opening downward and so y is positive for x between -1 and 1. I don't where you got the "0" in "0< x< 1". Was that a typo?

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What is the first derivative?

The first derivative is a mathematical concept that represents the rate of change or slope of a function at a specific point. It is written as f'(x) or dy/dx.

Why is the first derivative important?

The first derivative is important because it can tell us how a function is changing at any given point. It can also help us find maximum and minimum points, as well as the concavity of a function.

How do you calculate the first derivative?

The first derivative can be calculated using the power rule, product rule, quotient rule, or chain rule, depending on the function given. It is important to have a good understanding of basic calculus concepts and rules in order to calculate the first derivative accurately.

What is the difference between the first derivative and the second derivative?

The first derivative represents the rate of change of a function, while the second derivative represents the rate of change of the first derivative. In other words, the second derivative tells us how the slope of the function is changing at a specific point.

How can the first derivative be applied in real life?

The first derivative has many applications in real life, such as in physics to calculate velocity and acceleration, in economics to find maximum profit or minimum cost, and in biology to model growth and decay. It is a fundamental concept in understanding and analyzing various natural phenomena.

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