Q: 

A.  Implicitly find dy/dx of e^xy=8

B.  Now, solve e^xy=8 for y first, then take the derivative.  Compare your answers to A and B.

Answer:

A.  Remember, implicit differentiation is just the chain rule:  y is a function of x.  So, we need to use the product rule since we are multiplying two functions e^x times y:

Also remember, the derivative of x with respect to x is 1 and the derivative of y with respect to x is dy/dx:

Differentiate as follows:

e^xy + e^x(dy/dx) = 0

Now, solve for dy/dx:

e^x(dy/dx) = -e^xy

dy/dx = -e^xy/e^x

dy/dx = -y [final answer]

However, we may want to put the answer in terms of x.  If this is the case, we know from the original problem:

e^xy=8 which implies:

y = 8/e^x

So, dy/dx = -y = -8/e^x

B.  Solve for y first, then take the derivative:

We just solved for y above and got:

y = 8/e^x

We can rewrite this as:

y = 8*e^-x

Now, take the derivative (using the chain rule)

dy/dx = 8*e^-x*(-1)

dy/dx = -8*e^-x

Which can then be re-written as:

dy/dx = -8/e^x

Compare A and B?  Notice they are identical — which is good news.  Whether we solve for y first or differentiate implicitly, we still get the same answer for dy/dx.