We first find y in terms of x, not for differentiation but for a later reason. This is an example of implicit differentiation, where each y term is differentiated as if it were respect to x and then multiplied by dy/dx. The product rule is used to differentiate a non-differentiable product by splitting it into two differentiable terms, equating them to u and v, differentiating individually, and combining the results. Putting the left and right hand sides back together and multiplying by y gives the derivative of y in terms of only x.

1. We first find y in terms
of x.
This is not for the
purpose of
Differentiation, but for
a reason seen later.
Left Hand Side (while finding )
When differentiating a y
value with respect to y, you
may differentiate each y
This is an example term as if it were with
of Implicit respect to x, however, you
Differentiation. must then multiply the
derivative by .
Right Hand Side (while finding )
This is an example of the Product Rule:
You take a non-differentiable product, and split it up into two differentiable terms
(in this case 3x and ln x). Equate one term to u, and another to v (or equivalent),
and then differentiate u and v, individually, which results in ,
respectively. , in this instance, is equal to: .
The result to this equation is as follows…
Therefore:
Putting the Left and Right
hand sides back together
gives us the following:
Multiply both sides by y to
get a value of y on the right
hand side.
You can then substitute the
value of y in terms of x into
the equation.
Thus, we now have the
derivative of y in terms of
only x.