How can I find the derivatives of the implicitly set function y=y(x), set with the equation #xe^y + ye^x - e^(xy) = 0#? the answer I have is y' = ye^xy -e^y -ye^x/xe^y + e^x -xe^xy
2 Answers
Explanation:
Given:
Differentiate each term with respect to x:
The first term requires the use of the product rule:
Where
We need the chain rule for
Substituting into the product rule:
Substitute equation [2] into equation [1]:
The next term, also, requires the use of the product rule:
Where
Substituting into the product rule:
Substitute equation [3] into equation [1.1]:
If we let
But we shall need the product rule to compute
Reverse the substitution for u:
Use the distributive property:
Substitute equation [4] into equation [1.2] (remember to distribute the leading -1):
The derivative of 0 is 0:
Move all of the terms that do NOT contain
Factor out
Divide both sides by
Explanation:
notice that
this will give you
applying product rule
so
rearrange the equation,
y'=
alternatively,