# Help w/ Calc 1 derivative (simple)

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In summary, the student is struggling with an equation and is having trouble simplifying it. They say they got the simplification but don't remember how to do the flip trick.
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## Homework Statement

f(x) = (x) / (x + (c/x) )

## Homework Equations

Product rule and quotient rule

## The Attempt at a Solution

f'(x) = [ (x + (c/x)) - (x(1-(c/x^2)) ] / [ (s + (c/x)^2) ]
= [x + (c/x) - x + (c/x)] / [(x+(c/x)^2)]
= [2c/x] / [(x + (c/x)^2)]

That's where I'm stuck. I need to brush up on algebra, I know. If anyone can give me any good review sites, that'd be great.

I'm just doing problems in the book in my own time, and this is an odd number. The answer is:
[2cx] / [ (9x^2)+c)^2 ]

Was I even going along in the right direction? I also tried doing it as simply a product rule, where I just made it cx^-1, but that got messy when I tried to simplify.

You've got some typos in there, but if you meant your answer to be (2c/x)/(x+c/x)^2 then that looks correct. You should be able to simplify that to 2cx/(x^2+c)^2. You won't get 2cx/(9*x^2+c)^2. That doesn't look right at all.

There seems to be a problem with the answer. There shouldn't be a 9.

I'll continue after where you got stuck:

(2c/x)/(x + c/x)^2

= (2c/x)/[((x^2 + c)/x)^2] (Making the common denominator x of the denominator)
= (2c/x)/[((x^2 + c)^2)/x^2] (Splitting up the square between the numerator and denominator of the denominator)

Then if you remember dividing fractions, then you know that all it is is multiplying one fraction with the reciprocal of the other fraction. Do the same thing here and you'll get:

2cx/(x^2 + c)^2

Yeah, that's what I meant. I guess my finger slipped off the shift key.

I got the simplification. I can't believe I forgot the old flip trick to turn division into multiplication. Thanks to both of you.

## 1. What is a derivative in Calculus?

A derivative is a mathematical tool used to measure the rate of change of a function at a specific point. It is a fundamental concept in calculus that is used to find slopes of curves, determine maximum and minimum values, and solve optimization problems.

## 2. How do I find the derivative of a simple function in Calculus?

To find the derivative of a simple function, you can use the power rule, product rule, quotient rule, or chain rule. These rules provide a step-by-step method for finding the derivative of a function based on its algebraic form.

## 3. What is the difference between a derivative and a differential?

A derivative is a function that represents the rate of change of another function, while a differential is an infinitesimal change in a variable. In simpler terms, a derivative is a slope and a differential is a change in x or y.

## 4. How do I use the derivative to find the slope of a tangent line?

To find the slope of a tangent line using the derivative, you need to find the derivative of the function and then plug in the x-coordinate of the point where you want to find the slope. The resulting value is the slope of the tangent line at that point.

## 5. Can I use the derivative to find the maximum or minimum value of a function?

Yes, derivatives can be used to find the maximum and minimum values of a function. The maximum value occurs at the point where the derivative changes from positive to negative, while the minimum value occurs at the point where the derivative changes from negative to positive.

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