Complicated implicit differentiation

[cal.queen92]

Homework Statement

If (sqrt x) + (sqrt y) = 11 and f(9)=64 ---> find f '(9) by implicit differentiation

The Attempt at a Solution

I keep getting lost in my work here...

first, taking derivative of both sides:

d/dx ((sqrt x) + (sqrt y)) = d/dx (11)

obtaining: (1/2)(x)^(-1/2) + (1/2)(y)^(-1/2) * dy/dx = 0

Now, I want to keep y positive so:

(1/2)(y)^(-1/2) * dy/dx = -(1/2)(x)^(-1/2)

So if I solve for dy/dx:

dy/dx = (-1/(sqrt x)) / (1/(sqrt y)) which means: dy/dx = (-1/(sqrt x) * ((sqrt y)/1)

giving: dy/dx = -(sqrt y)/ (sqrt x) as the derivative.

However, I don't know how to use the other information provided! I am very stuck... If anyone has any ideas, thanks!

 

[Mark44]

Homework Statement

If (sqrt x) + (sqrt y) = 11 and f(9)=64 ---> find f '(9) by implicit differentiation

The Attempt at a Solution

I keep getting lost in my work here...

first, taking derivative of both sides:

d/dx ((sqrt x) + (sqrt y)) = d/dx (11)

obtaining: (1/2)(x)^(-1/2) + (1/2)(y)^(-1/2) * dy/dx = 0

Now, I want to keep y positive so:

(1/2)(y)^(-1/2) * dy/dx = -(1/2)(x)^(-1/2)

So if I solve for dy/dx:

dy/dx = (-1/(sqrt x)) / (1/(sqrt y)) which means: dy/dx = (-1/(sqrt x) * ((sqrt y)/1)

giving: dy/dx = -(sqrt y)/ (sqrt x) as the derivative.

However, I don't know how to use the other information provided! I am very stuck... If anyone has any ideas, thanks!

Your derivative is correct. The information you didn't use is f(9)=64. Here f is the implicit function defined by the equation x^1/2 + y^1/2 = 11.

f(9) = 64 means that the point (9, 64) is on the graph of the implicit function. What you're supposed to do to finish this problem is to find dy/dx when x = 9 and y = 64.

 

[cal.queen92]

I see! That's great! so then that gives an answer of -8/3. Perfect! Thank you.