Homework Help: How to find differencial by using implicit functions?

1. Mar 18, 2013

munkhuu1

1. The problem statement, all variables and given/known data
R=1/(.55/c+.45/h)
find partial equations respect to c. and respect to h
use implicit function differentiation of the reciprocal of R to answer
what is the differential change in R when c=20 h=30 and c changes to 21

2. Relevant equations
is there any way to make R easier?
i said that R=ch/(.55h+.45c) which was the best i could do.
Is there any other way to make R easier?

3. The attempt at a solution
i got the partial equations if the R=ch/(.55h+.45c) is right.
but im not sure how to use implicit or what implicit is. i just found it normally.
WHen it says what is the differential change in R when c=20 and h=3 and c changes to 21. do i just substitude them to the partial differentials and add them?

2. Mar 18, 2013

HallsofIvy

Yes, R= ch/(.55h+ .45c) is correct and about as simple as it gets. I notice that the problem asks you to "use implicit function differentiation of the reciprocal of R. That is, of course, 0.55h+ 0.45c= chR. Differentiate both sides of that with respect to h to find $\partial R/\partial h$. (Surely you remember "implicit differentiation" from Calculus I?)

In general the "differential" of a function, f(h,c), is
$$df= \frac{\partial f}{\partial h}dh+ \frac{\partial f}{\partial c}dc$$

But notice that, in this problem, only c changes.

3. Mar 18, 2013

munkhuu1

thank you.
just one more thing.
could you walk me through implicit differentiation on this problem?
i start off when i find dR/dh. i got .55+0=crDr/dh ? is this right or am i missing something?

4. Mar 18, 2013

economicstyro

To use implicit differentiation, you need to first present the given equation in the form F(R,c,h) by isolating the three variables into one side. So in our case, F(R,c,h)=R-$\frac{ch}{0.55h+0.45c}$=0. And according to the Implicit Differentiation Rule, ∂R/∂c=-(∂F/∂c)/(∂F/∂h) (notice the negative sign!!), where ∂F/∂c=[h*(0.55h+0.45c)-ch*(0.45)]/[0.55h+0.45c]^2 (using the quotient rule and considering h and R constants, we have differentiated F with respect to c); and ∂F/∂R=1. Proceed in a similar fashion and we will get ∂R/∂h.