Derivative of g(h(t), t) with respect to h

[jstock23]

Yoooooooooooo,

How would the derivative of a function with respect to another function work?

For example:

h(t) = t^2
g(h,t) = ht = t^3

dg/dh = (d/dh)ht

What I think:

maybe
dg/dh = t

chain rule
dg/dt = (dg/dh)(dh/dt) + dt/dt
then solve for dg/dh

or maybe
dg/dt = (dg/dh)(dh/dt)
then solve for dg/dh

or maybe
dg/dt = (dh/dt)t + (dt/dt)h
so it's impossible

Having some trouble deciding. Thx a bunch, I've been having a huge problem with this.

 

[Stephen Tashi]

Try
$$\frac {dg}{dt} = \frac{\partial g}{\partial h} \frac{dh}{dt} + \frac{\partial g}{\partial t} \frac{dt}{dt}$$

$$= (t)(2t) + (h)(1) = 2t^2 + (t^2)(1) = 3t^2$$

Having said that, when I read books on classical mechanics where the derivatives are denoted with dots, it always looks like they aren't following that rule exactly.

 

[jstock23]

Ooh, I messed up the chain rule hehe

How can you take a partial derivative with respect to h, holding t constant, while this implies that h will also be constant, as it is a function of t?

I trust what you said, and the chain rule, it still just isn't that convincing because of the dependent nature of h.

Thx again

 

[Stephen Tashi]

I trust what you said, and the chain rule, it still just isn't that convincing because of the dependent nature of h.

Maybe that's the same trouble I have when I read those classical mechanics books!

I think the way to derive that formula involves considering g(h,t) to be a surface over the (h,t) plane. Then consider a path defined by formula (t^2,t) that passes through a point (h,t). Let dg/dt be the rate of change in g as seen by someone moving along that path. Being on the path introduces a relation between h and t. The path has a direction and a speed (with respect to a change in t). As I recall, analyzing all that (which is usually done in terms of vectors) gives the formula.

 

[jstock23]

I think the way to derive that formula involves considering g(h,t) to be a surface over the (h,t) plane.

cool, that spatial example made it all make sense hehe :)