Finding d2y/dz2 for Calculating Derivatives with Variable Change

[mattlorig]

Okay, I feel really dumb, because this seems like something I should know, but I don't. So, here's my question.

Suppose I have d2y/dx2, x = x(z), and z = z(x)
How do I find d2y/dz2?
Here's what I did:

dy/dz = dy/dx * dx/dz
d2y/dz2 = d/dz ( dy/ dz ) = d/dz (dy/dx * dx/dz)
= d2y/dx2 * (dx/dz)^2 + dy/dx * d2x/dz2

Is that OK? Is there an easier way to do this?

 

[dextercioby]

Nope.2 times chain rule required by the implicite dependence upon a variable...

Daniel.

 

[thepatientmental]

Don't feel dumb for not knowing this! Calculating derivatives with variable change can be confusing at first, but with practice and understanding the concepts, it will become easier.

Your method for finding d2y/dz2 is correct. Another way to approach it is by using the chain rule. Remember that d2y/dz2 is the second derivative of y with respect to z, so we can rewrite it as d/dz (dy/dz). Then, using the chain rule, we can write it as d/dx (dy/dz) * dz/dz, since dz/dz is simply 1. This simplifies to d/dx (dy/dz), which is equivalent to d2y/dx2. So, another way to find d2y/dz2 would be to first find d2y/dx2 and then substitute in the values for dx/dz and d2x/dz2, as you did in your method.

Overall, both methods are correct and it's just a matter of personal preference which one you choose to use. Keep practicing and you'll become more comfortable with finding derivatives with variable change!