How to find differencial by using implicit functions?

[munkhuu1]

Homework Statement

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

Homework 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?

The Attempt at a Solution

i got the partial equations if the R=ch/(.55h+.45c) is right.
but I am 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?

 

[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.

 

[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?

 

[economicstyro]

Hello munkhuu, I believe this link would be tremendously helpful in solving your problem: http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx

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.

As for part b), in order to find the marginal effect of c on R (notice that h remains unchanged), we just need to multiply ∂R/∂c with ∂c to get rid of the denominator, and plug in (c=21, h-30, ∂c (change in c)=1). I hope my approach is right. Good luck :)