Find the second derivative using the quotient rule

[DieCommie]

Homework Statement

$$\frac{d^2}{dz^2} [ (z^2) \(\frac{-(z-\frac{1}{z})^2}{20+8(z+\frac{1}{z})} \frac{1}{iz}]$$

Homework Equations

The Attempt at a Solution

I have to do this by hand no calculators, CAS, or tables. I started by just expanding things and using the quotient rule, but it got real messy real fast. Any ideas?

 

[rs1n]

It is often easier to simplify first, and then differentiate. Let's see if we can rewrite your product of fractions:

$$z^2\cdot \frac{-(z-\frac{1}{z})^2}{20+8(z+\frac{1}{z})} \cdot \frac{1}{iz}<br /> = \frac{z^2[-(z^2-2+\frac{1}{z^2})]}{20+8(z+\frac{1}{z})} \cdot \frac{1}{iz}<br /> = \frac{-(z^4 + 2z^2 + 1)}{[20+8(z+\frac{1}{z})]z} \cdot \frac{1}{i}<br /> = -\frac{z^4 + 2z^2 + 1}{20z+8z^2+8} \cdot \frac{1}{i}<br /> = -\frac{z^4+2z+1}{8z^2+20z+8} \cdot (-i)$$

which finally simplifies to

$$i \cdot \frac{z^4 + 2z + 1}{8z^2+20z+8} =<br /> \frac{i}{4} \cdot \frac{z^4+2z^2+1}{2z^2+5z+2}$$

You can stop here and use the quotient rule for the first derivative; then differentiate the result to obtain the second derivative.

You may find it easier to write this fraction as a product (I personally consider the product rule to be cleaner and faster than the quotient rule, though they are equivalent in many respects).$$\frac{i}{4} \cdot \frac{z^4+2z^2+1}{2z^2+5z+2} =<br /> \frac{i}{4} (z^4+2z+1) \cdot (2z^2+5z+2)^{-1}$$

Using this form, just differentiate using the product rule, and don't simplify the result (because it makes finding the second derivative easier; we just apply the product rule again).