Implicit Differentiation

In calculus, when applications are necessary for problems of related rates and the like, Implicit Differentiation is one of the areas of calculus that has to be mastered.

As the name implies, Implicit means indirect differentiation.  Luckily, it is very methodical and can be easily understood by following simple guidelines.


To do Implicit derivatives, one must take the derivative of a given equation written with both x and y variables.

i.e.  5 = xy+3x+1

Here, we take the derivative of the entire function, and as we do, multiply the derivative of y with .

So, since the first term involves a product of x and y, we have to use the product rule, so we write:

1*y+x(3y())+6x+0=0

then, we move everything but the term containing to the other side of the equation, so:

x(3y())=-y-6x

Laslty, we divide by x(3y) on both sides to solve for and we are done:


=(-y-6x)/(x(3y))