Survey of Calculus – Section 3.6 – Implicit Differentiation and... explicitly implicitly

Survey of Calculus – Section 3.6 – Implicit Differentiation and Related Rates
Until now all the functions you have seen in this class have been written explicitly. That just means that y has
been isolated. Now we will learn how to find the derivative when a function is written implicitly. In other
words, how do we find the derivative if it is difficult or impossible to isolate y?
Steps to evaluate a derivative implicitly
1) Differentiate both sides of the equation with respect to x. (Remember that y is a function of x, so you will
have to use the chain rule.)
2) Collect all terms involving
3) Factor out the
on one side, and all others on the other side.
and solve for it by dividing.
Of course, your variables do not have to be y and x as seen in the next examples.
Implicit differentiation is used to solve related rate problems. These are problems that show how fast one
quantity is changing relative to another. Our variable will usually be t to represent time. When setting these
problems up remember that the rate of change of a variable is given by its derivative with respect to time.