MTH 151 Midterm: Term Test 2 - Fall 2017

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Miami University
MTH 151
Calculus I
Fall 2017
Term Test 2
Prof: Olga Brezhneva
Exam Guide
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Chapter 3 Differentiation Rules
3.5 Implicit Differentiation
1. Definition
Implicit Function: An implicit function is a function that is defined implicitly by an implicit equation, by
associating one of the variables (the value) with the others (the arguments).
EX:
2. Implicit Differentiation
For implicit function, we don’t need to solve for y in term of x in order to find y’. What we can do it
differentiate both side of the function with respect to x, then solving the resulting question for y’.
Example:
Suppose that , ( y implicitly as a function of x ). If we want to find
, we need to
differentiate the entire equation with respect to x. Remember that y is also a function of x.















The above process to find 
 is called implicit differentiation.
Past Exam Questions:
1. Given , express 
 in terms of x and y. Evaluate at ( 0, 1 ).
Solution:
Differentiate both side with respect to x:



(0)
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











At the point (0,1), 


2. Given , find
.
Solution:


















Then we have:






On the left side:











On the right side:













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