How do you differentiate #e^sqrt(xy)-sqrt(xy)=8#?

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Answer 1 · 2015-12-20T16:31:11

See explanation.

#dy/dx = -c/(x^2)#

where #c# is the square of the positive root of #e^t-t-8 = 0#

Let #t = sqrt(xy)#

Then #e^t - t = 8#

This has two roots #t_1 ~~ 2.33559#, #t_2 ~~ -7.999664#

Numerical approximations for #t_1# and #t_2# can be found using Newton's method or similar.

We can discard #t_2# since #t = sqrt(xy)# is the non-negative square root.

Let #c = t_1^2#

Then the original equation simplifies to:

#xy = c#

So #y = c/x# and #(dy)/(dx) = -c/(x^2)#