Question

Having trouble with quotient rule - Calc help please.

(x^2 + 1)^4 / x^4

Answer 1

I have to derive that

Answer 2

The answer is supposed to be [4(x^2-1)(x^2+1)]/x^5 which is not what I get

Answer 3

sorry the answer is [4(x^2-1)(x^2+1)^3]/x^5

Answer 4

First write it in the equation editor to get a better look at it:\[\frac{ (x^2+1)^4 }{ x^4 }=\left( \frac{ x^2+1 }{ x } \right)^4\]
Now you can also use the Chain Rule, because we have the 4th power as a second step.

The derivative is:\[4\left( \frac{ x^2+1 }{ x } \right)^3 \cdot \frac{ x \cdot 2x - 1 \cdot(x^2+1) }{ x^2 }=4\left( \frac{ x^2+1 }{ x } \right)^3 \cdot \frac{ x^2- 1}{ x^2 }\]

Answer 5

Can I use chain rule where there is a fraction?

Answer 6

Although we're done differentiating, we could still simplify:\[\frac{ 4(x^2+1)^3(x^2-1) }{ x^5 }\]

Answer 7

@pottersheep: as long as there is a chain, you can (have to) use the Chain Rule!

The chain here is:

1. u=(x^2+1)/x
2. y=u^4

Answer 8

Ah ok

Answer 9

Thank you for your explanation

Answer 10

You could have done it without Chain Rule, if you leave the original function as it was:\[\left( \frac{ (x^2 + 1)^4 }{ x^4 }\right)'=\frac{ x^4 \cdot 4(x^2+1)^3 \cdot 2x-(x^2+1)^4 \cdot 4x^3 }{ x^8 }\] (still used Chain Rule - see factor 2x!)

This now also has to be simplified (begin with dividing everything by x³):\[\frac{ 8x^2(x^2+1)^3-4(x^2+1)^4 }{ x^5 }=\]\[\frac{ 4(x^2+1)^3(2x^2-(x^2+1)) }{ x^5 }=\frac{ 4(x^2+1)^3(x^2-1) }{ x^5 }\]So we come to the same answer, eventually.
You decide which method is easier!

Answer 11

OHhhhh that's what I did before! I see my mistake now! :)

Answer 12

Thanks again!

Answer 13

yw!