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
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dy
on one side, and all others on the other side.
dx
dy
and solve for it by dividing.
dx
Of course, your variables do not have to be y and x as seen in the next examples.
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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.
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