# 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,