# What is the first derivative and critical numbers of #y= (x^4)/(x^4-1)#?

Critical points are at

This derivative can of course be computed by using the quotient rule. However, that tends to be somewhat cumbersome, so we can be clever. If we use a bit of algebraic long division, we can get away with a simple chain rule application.

Resubstitute:

To solve for the critical points, we simply set the derivative expression equal to 0 and solve for x, and also find any x's where the function is undefined:

When we have an expression of something divided by something else equals 0, we can deduce that the top expression must be 0, since if the bottom were to be 0, the value would be undefined.

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To find the first derivative of ( y = \frac{x^4}{x^4 - 1} ), you can use the quotient rule. The first derivative, ( y' ), is:

[ y' = \frac{(x^4 - 1)(4x^3) - (x^4)(4x^3)}{(x^4 - 1)^2} ]

To find the critical numbers, set ( y' ) equal to zero and solve for ( x ). Then check for any points where the derivative is undefined (where the denominator equals zero).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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