Simplifying a Function Prior to Finding a Derivative

In summary, the book simplifies a problem before finding the derivative sometimes, but sometimes the book doesn't simplify it and you have to do it yourself.
  • #1
Drakkith
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Okay guys, this is driving me absolutely nuts.
I'm working on finding derivatives using the product and quotient rules and the book will sometimes simplify the problem before finding the derivative but sometimes wont and I don't understand why.

For example: The function y = (v3-2v√v)/v
The book simplifies it to v2-2v½ and then gets 2v-v as the derivative without even using the product or quotient rules.

Then on this problem: f(x) = x/[x+(c/x)] it doesn't simplify first. I started off by simplifying it as follows:
Original function: f(x) = x/[x+(c/x)]
Making the bottom all one fraction: x/[(x2+c)/x]
Bottom and top cancel out: 1/(x2+c).
But using the quotient rule on this gives me an answer of 2x/(x2+c)2, which is wrong. The answer is actually 2cx/(x2+c)2, which the book gets by using the quotient rule without simplifying first.

When is it okay to simplify before finding the derivative and when isn't it?
 
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  • #2
Drakkith said:
Bottom and top cancel out
They don't. You divide the numerator by x but multiply the denominator with x.

Drakkith said:
I'm working on finding derivatives using the product and quotient rules and the book will sometimes simplify the problem before finding the derivative but sometimes wont and I don't understand why.
It probably chooses the most convenient option. A proper simplification does not change the function, you can always do it if you like.
 
  • #3
Dang it, I didn't switch the fraction around when I was doing the multiplication. Sigh... it's been like this all day today...

So x/[(x2+c)/x] is equal to x2/(x2+c)
So the derivative is: (x2+c)(2x) - (x2)(2x) / (x2+c)2
Which then becomes: 2x3+2cx-2x3 / (x2+c)2
Or: 2cx/(x2+c)2, which is the correct answer...

Thanks Mfb.
 

FAQ: Simplifying a Function Prior to Finding a Derivative

What is simplifying a function?

Simplifying a function involves reducing the function to its simplest form by combining like terms, factoring, and applying other algebraic techniques.

Why is simplifying a function important prior to finding a derivative?

Simplifying a function prior to finding a derivative can make the process easier and more efficient. It can also help to identify patterns and relationships within the function.

What are some common techniques for simplifying a function?

Common techniques for simplifying a function include combining like terms, factoring, using the distributive property, and applying the rules of exponents.

What are some common mistakes to avoid when simplifying a function?

Some common mistakes to avoid when simplifying a function include forgetting to distribute a negative sign, incorrectly combining terms, and making errors with exponents.

How can simplifying a function help in understanding the derivative?

Simplifying a function can help to identify the behavior of the function and reveal any symmetries or patterns that can inform the understanding of the derivative. It can also make it easier to apply derivative rules and identify the critical points of the function.

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