Derivative of y = x (1 - x^2)^1/2, is this correct?

  • Thread starter Leonidas
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In summary, the conversation discussed finding the derivative of y=x(1-x^2)^1/2 using the product rule and chain rule. The correct answer was obtained, but there was a small mistake in applying the chain rule. The correct derivative is y=(-x^2/√(1-x^2))+(1-x^2)^1/2.
  • #1
Leonidas
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I keep on getting a weird answer when i take the dy/dx for this...
y=x(1-x^2)^1/2

i got x(1/x^2)^(-1/2)*-2x + (1-x^2)^(1/2)

... did i do that correctly?
 
Last edited:
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  • #2
[tex] y=x(1-x^2)^\frac{1}{2} [/tex]

Use product rule/chain rule:

[tex]
y=(x)(\frac{1}{2})(\frac{1}{\sqrt{1-x^2}})(-2x) + (1-x^2)^\frac{1}{2}
[/tex]

Slightly more simplified:

[tex]
y=(\frac{-x^2}{\sqrt{1-x^2}}) + (1-x^2)^\frac{1}{2}
[/tex]

Pretty sure that's the answer...if so you aren't using the chain rule correctly. Double check how you get the derivitive of [tex] (1-x^2)^\frac{1}{2} [/tex]
 
  • #3
DUH. *slaps self on forhead* ... heh... forgot a single step and it screwed me up (of course).

Thanks.
 
Last edited:

1. What is the formula for the derivative of y = x (1 - x^2)^1/2?

The formula for the derivative of y = x (1 - x^2)^1/2 is given by:
y' = (1 - x^2)^1/2 - x(1/2)(2x)(1 - x^2)^-1/2
Simplifying, we get y' = (1 - x^2)^1/2 - x^2(1 - x^2)^-1/2.

2. How do you find the derivative of a function with a square root?

To find the derivative of a function with a square root, we use the power rule and chain rule.
First, we rewrite the function in the form (u^n)^m, where u is the inner function and m is the power of the entire function.
Then, we use the chain rule to find du/dx.
Finally, we apply the power rule by multiplying the derivative of the inner function (du/dx) with the power (m) and subtracting 1 from the power (m-1).

3. Can the derivative of y = x (1 - x^2)^1/2 be simplified further?

Yes, the derivative of y = x (1 - x^2)^1/2 can be simplified further by factoring out a common term.
We can rewrite the derivative as y' = (1 - x^2)^1/2 (1 - x)

4. What does the derivative of y = x (1 - x^2)^1/2 represent?

The derivative of y = x (1 - x^2)^1/2 represents the rate of change of the original function at any given point. In other words, it tells us how fast the function is changing at a specific value of x.

5. Is the derivative of y = x (1 - x^2)^1/2 defined for all values of x?

Yes, the derivative of y = x (1 - x^2)^1/2 is defined for all values of x, as long as the function is continuous and differentiable at those points. However, there may be some points where the derivative is undefined due to discontinuities or sharp turns in the function.

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