# Thread: Need help to find the first order partial derivatives

1. ## Need help to find the first order partial derivatives

This is the given problem I have:
Use the Chain Rule (or the Invariance Property of diﬀerentials) to ﬁnd the ﬁrst order partial derivatives (i.e. ∂g/∂s and ∂g/∂t)
of g(s,t) := f(s,u(s,t),v(s,t)), where
u(s,t) := st,
v(s,t) := s + t.
The answer should be expressed in terms of s and t only.

I'm completely stuck on this problem, due to the Equation itself. Dont really know how to do it, anyone that can help me?

2. ## Re: Need help to find the first order partial derivatives

Well, you use the phrase "Chain Rule" so apparently you know what that is! Here, g, a function of s and t, is equal to f, a function of the three variables, s, u, and v, with u and v themselves functions of s and t.

By the "Chain Rule", $\frac{\partial g}{\partial s}= \frac{\partial f}{\partial s}+ \frac{\partial f}{\partial u}\frac{\partial u}{\partial s}+ \frac{\partial f}{\partial v}\frac{\partial v}{\partial s}$
(Unfortunate notation but I don't know any better! The first $\frac{\partial f}{\partial s}$ is just the derivative of f with respect to s, ignoring u and v and their dependence on s)

This is $\frac{\partial g}{\partial s}= \frac{\partial f}{\partial s}+ \frac{\partial f}{\partial u}(t)+ \frac{\partial f}{\partial v}(1)$.
Do the partial derivative of g with respect to t in the same way.

You can't do more than this without knowing what "f" is.