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What is the derivative of #y = ((x^2 + 5) / (x^2 - 5))^5#?

Answer 1

Luke Alvarez

#dy/dx= (-100x(x^2 + 5)^4)/(x^2 - 5)^6#

Use the chain rule.

Let #y = u^5# and #u = (x^2 + 5)/(x^2 -5)#.

#dy/dx= 5u^4 * (2x(x^2 - 5) - (2x(x^2 + 5)))/(x^2 - 5)^2#

#dy/dx = 5u^4 * (2x^3 - 10x - 2x^3 - 10x)/(x^2 -5)^2#

#dy/dx = 5u^4 * (-20x)/(x^2 -5)^2#

#dy/dx= (-100x((x^2 + 5)/(x^2 -5))^4)/(x^2 -5)^2#

#dy/dx= (-100x(x^2 + 5)^4)/(x^2 - 5)^6#

Hopefully this helps!

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Answer 2

Paisley Johnson

To find the derivative of the given function, y = ((x^2 + 5) / (x^2 - 5))^5, you can use the chain rule along with the quotient rule. After applying these rules, simplify the expression to find the derivative. The derivative is given by:

y' = 5 * ((x^2 + 5) / (x^2 - 5))^4 * ((2x(x^2 - 5) - 2x(x^2 + 5)) / (x^2 - 5)^2)

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Answer from HIX Tutor

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.