# Multivariable Chain rule for higher order derivatives

1. May 9, 2013

### Sunfire

Hello,

Given is the function

f = f(a,b,t), where a=a(b) and b = b(t). Need to express first and second order derivatives.

$\frac{\partial f}{\partial a}$ and $\frac{\partial f}{\partial b}$ should be just that, nothing more to it here, correct?

But

$\frac{df}{dt}$ = $\frac{\partial f}{\partial a}$ $\frac{da}{db}$ $\frac{db}{dt}$ + $\frac{\partial f}{\partial b}$ $\frac{db}{dt}$ + $\frac{\partial f}{\partial t}$, by the chain rule, correct?

I need to express $\frac{\partial f}{\partial t}$, but the above chain rule puts the total derivative $\frac{df}{dt}$ in the expression and it gets messy. I mean, how do I express

$\frac{\partial f}{\partial t}$?

Then I need also $\frac{\partial^2 f}{\partial t^2}$, $\frac{\partial^2 f}{\partial a^2}$ and $\frac{\partial^2 f}{\partial b^2}$.

Anyone well versed in partial derivatives?

2. May 12, 2013

### Sunfire

Let me try to clear it up a bit

Let f = f(a,b,t) where a=a(b(t)), b=b(t)

Then

$\frac{df}{dt}$=$\frac{∂f}{∂a}$$\frac{da}{db}$$\frac{db}{dt}$ + $\frac{∂f}{∂b}$$\frac{db}{dt}$ + $\frac{∂f}{∂t}$, correct?

How does one express

$\frac{d^2f}{dt^2}$=?

3. May 13, 2013

### Sunfire

Okay, Let me simplify this to

Let f = f(a,b) where a=a(b)

Then

$\frac{df}{db}=\frac{∂f}{∂a}\frac{da}{db} + \frac{∂f}{∂b}$, correct?

How does one express

$\frac{d^2f}{db^2}$=?