Finding the First Derivative of a Complicated Expression

In summary, the conversation is about finding the second derivative of a function using the quotient rule or product rule. The original function is simplified incorrectly and the correct steps are provided to simplify it. The correct answer is given and it is suggested to use the product rule to differentiate the first derivative.
  • #1
frosty8688
126
0
1. Simplify the derivative



2. [itex] y = \frac{\sqrt{1-x^{2}}}{x}[/itex]



3. [itex] f'(x) = \frac{x*\frac{-2x}{2\sqrt{1-x^{2}}}-\sqrt{1-x^{2}}}{x^{2}}=\frac{-x^{2}-\sqrt{1-x^{2}}}{x^{2}\sqrt{1-x^{2}}}[/itex] I just don't know how to simplify this further.
 
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  • #2
frosty8688 said:
1. Simplify the derivative

2. [itex] y = \frac{\sqrt{1-x^{2}}}{x}[/itex]

3. [itex] f'(x) = \frac{x*\frac{-2x}{2\sqrt{1-x^{2}}}-\sqrt{1-x^{2}}}{x^{2}}=\frac{-x^{2}-\sqrt{1-x^{2}}}{x^{2}\sqrt{1-x^{2}}}[/itex] I just don't know how to simplify this further.
You simplified ##\displaystyle \ \frac{\displaystyle x\frac{-2x}{2\sqrt{1-x^{2}}}-\sqrt{1-x^{2}}}{x^{2}} \ ## incorrectly.
 
  • #3
So it would be -x[itex]^{2}[/itex] over the square root. So how do I find the second derivative of [itex]\frac{\frac{-x^{2}}{\sqrt{1-x^{2}}}-\sqrt{1-x^{2}}}{x^{2}}[/itex] Do I use the product rule or quotient rule. Which is easier?
 
Last edited:
  • #4
Multiply by ##\displaystyle \ \frac{\sqrt{1-x^2}}{\sqrt{1-x^2}} \ .##
 
  • #5
SammyS said:
Multiply by ##\displaystyle \ \frac{\sqrt{1-x^2}}{\sqrt{1-x^2}} \ .##

I already did that and came out with [itex] \frac{-1}{x^{2}\sqrt{1-x^{2}}}[/itex]
 
  • #6
How do I take the second derivative?
 
  • #7
frosty8688 said:
How do I take the second derivative?
You can use the quotient rule. To differentiate the denominator use the product rule. Alternatively, rewrite it as a product of two terms with negative exponents and just use the product rule.
 
  • #8
Here is what I have [itex]\frac{\frac{-x}{\sqrt{1-x^{2}}}-\frac{x^{3}}{(1-x^{2})^{3/2}}} x^{2} + \frac{2}{x^{3}}[/itex] The squared on the top is supposed to go with the x on the bottom.
 
  • #9
frosty8688 said:
Here is what I have [itex]\frac{\frac{-x}{\sqrt{1-x^{2}}}-\frac{x^{3}}{(1-x^{2})^{3/2}}} x^{2} + \frac{2}{x^{3}}[/itex] The squared on the top is supposed to go with the x on the bottom.
You mean [itex]\displaystyle \ \ \frac{\displaystyle \frac{-x}{\sqrt{1-x^{2}}}-\frac{x^{3}}{(1-x^{2})^{3/2}}} {x^{2}} + \frac{2}{x^{3}}\ \ ?[/itex]

I don't see how that can possibly be correct.

Please show some steps.
 
  • #10
SammyS said:
You mean [itex]\displaystyle \ \ \frac{\displaystyle \frac{-x}{\sqrt{1-x^{2}}}-\frac{x^{3}}{(1-x^{2})^{3/2}}} {x^{2}} + \frac{2}{x^{3}}\ \ ?[/itex]

I don't see how that can possibly be correct.

Please show some steps.

Here is how I got that answer [itex]\frac{\frac{-2x^{3}}{2(1-x^{2})^{3/2}}}+\frac{2x}{2\sqrt{1-x^{2}}}-\frac{2x}{\sqrt{1-x^{2}}}{x^{2}}-2\frac{\frac{-x^{2}}{\sqrt{1-x^{2}}}}-\sqrt{1-x^{2}}{x^{3}} = \frac{\frac{-2x^{3}}{2(1-x^{2})^{3/2}}}-\frac{x}{\sqrt{1-x^{2}}}{x^{2}}-2\frac{\frac{-x^{2}}{\sqrt{1-x^{2}}}}-\sqrt{1-x^{2}}{x^{3}} = \frac{\frac{-x}{\sqrt{1-x^{2}}}}-\frac{x^{3}}{(1-x^{2})^{3/2}}{x^{2}}-2\frac{\frac{-x^{2}}{\sqrt{1-x^{2}}}}-\sqrt{1-x^{2}}{x^{3}}=\frac{\frac{-x}{\sqrt{1-x^{2}}}}-\frac{x^{3}}{(1-x^{2})^{3/2}}{x^{2}}+\frac{2}{x^{3}}[/itex] That is how I got the answer. The + sign in the first part should be in the numerator and the x^2 should be in the denominator. Same thing with the sign in the second half of the first part and the sqrt should be in the numerator with the x^3 on the bottom, same thing in the second part and third part and last part. Sorry it got messed up.
 
  • #11
It doesn't need to be that complicated. Write the first derivative as ##-x^{-2}(1-x^2)^{-\frac 12}## and differentiate using the product rule. Hint: every term in the answer ought to have a factor like ##(1-x^2)^{n-\frac 12}##, some integer n. I'd guess that's how SammyS knew your answer could not be right.
 

What does it mean to "simplify" an expression?

Simplifying an expression means to rewrite it in a way that is easier to understand and work with, without changing the overall value of the expression.

Why is it important to simplify expressions?

Simplifying expressions can make them easier to solve, manipulate, and interpret. It can also help identify patterns and relationships within the expression.

What are some common techniques for simplifying expressions?

Some common techniques for simplifying expressions include combining like terms, using the distributive property, and factoring out common factors.

How do I know when an expression is fully simplified?

An expression is considered fully simplified when all possible simplifications have been made and no further changes can be made without altering the value of the expression.

Can simplifying an expression change its value?

No, simplifying an expression should not change its value. The goal of simplifying is to make the expression easier to work with, without changing its overall value.

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