implicit differentiation (Mathematics)

Question

Find dy/dx for e^(2x+3y) = x^2 -ln (xy^3)

Answer 1

e^(2x+3y) = x² - ln(xy³)

 

Nicolee, the first thing I would do would be to expand the "ln" using logarithm properties to keep from doing a complicated chain rule.

 

e^(2x+3y) = x² - (lnx + 3lny)

 

e^(2x+3y) = x² - lnx -3lny

 

No differentiate each side

 

deriv of e^anything is e^anything * deriv of "anything".  So the left side becomes

 

e^{(2x+3y)*[2 + 3 (dy/dx)]}  

 

diff the right side and get 2x - (1/x) - 3(1/y)(dy/dx)

 

distribute on the left

e^(2x+3y)(2) + e^(2x+3y)[3(dy/dx)] = 2x - (1/x) -(3/y)(dy/dx)

 

Now put all the (dy/dx)'s on the left.   Put everything else on the right.

 

Factor a (dy/dx) out of every term on the left

 

And divide both sides by the stuff on the left in the parentheses. 

 

Sorry - gotta run or I would show these steps...