IMPLICIT DIFFERENTIATION
IMPLICIT DIFFERENTIATION
Implicit vs. Explicit Functions
Example of explicit form: [pic]
❖ y is isolated … can be written f(x) =
❖ The right side is all in terms of x
❖ Differentiation the “normal” way
❖ But also could have originally been written as[pic], etc.
Example of two other equations in implicit form:
Implicit Form Explicit Form Derivative
[pic] [pic] [pic]
[pic] ??? ????
So … in order to differentiate … perhaps another technique would be useful … one that would leave the equation in IMPLICIT FORM … namely …
“IMPLICIT DIFFERENTIATION”
EXAMPLE 1:
a) [pic]… because the dx matches the x3
Thinking about it from the chain rule point of view:
[pic]
b) [pic]… because the dx does not match the y3
Thinking about it from the chain rule point of view:
[pic]
c) [pic]… sum/difference rule, then implicit differentiation (think chain rule)
d) [pic] … use implicit differentiation with the product rule…
[pic]
General Guidelines for Implicit Differentiation
1. Differentiate both sides of an equation with respect to x.
2. Collect [pic]terms on the left side, and move the others to the right side.
3. Factor out the[pic] from the left side terms.
4. Solve for[pic], by dividing both sides by the remaining terms that [pic] was factored out of.
EXAMPLE 2:
Find [pic]for [pic]
[pic]
[pic]
[pic]
EXAMPLE 3:
Rewrite each as a differentiable function:
a) [pic]
This is the equation of a single point therefore is not differentiable at all … so drop it right here!
b) [pic]
This is a circle with radius = 1. Isolate y …
[pic] … [pic]
… You can now use “normal” differentiation …
[pic]
Compare to the use of implicit differentiation.
[pic] … [pic] … [pic]
c) [pic]
This is a parabola which is symmetric with respect to the x-axis … vertex (1, 0) … and opens to the left …
Isolate y … and differentiate …
[pic] … [pic]
…now repeat with implicit differentiation, and compare…
[pic] … [pic]
Note: Since examples b and c, are not really functions when in implicit form … notice how the derivative depends on which branch of the relation you are using. Hence, the implicit form already has this taken care of, since it uses a y in the derivative.
EXAMPLE 4:
Determine the slope of a tangent line at the given point:
[pic] … [pic]
…Use implicit …
[pic]
Solve for [pic] … [pic] … at …[pic]
… [pic] …
… Equation: [pic]
Note: Isolate y, then try this the “normal” way later in the privacy of you own home ... See which makes better sense to use.
EXAMPLE 5:
Find the slope of the Lemniscate at (3, 1):
[pic]…use implicit, chain and product rules
[pic]
… distribute …
[pic]
… Isolate [pic] terms … and factor …
[pic]
… Solve for [pic] …
[pic]
… plug in (3, 1) …
[pic]
EXAMPLE 6:
Find [pic] for: [pic]
… Use implicit differentiation …
[pic]
… Solve for [pic] …
[pic]
…Set up a triangle to interpret the original equation, and then use it to rewrite answer in terms of x.
The Pythagorean Theorem gives you … ADJ? =[pic]
[pic]
[pic]
EXAMPLE 7:
Find [pic] for [pic]***
… that is, find the second derivative …
… so use “implicit” to find [pic] first …
[pic] … [pic]
… now use the quotient rule … and replace[pic]with [pic]
[pic]
…simplify by replacing [pic]***
[pic][pic]
EXAMPLE 8
Find the tangent line to the graph given by the equation:
[pic]at [pic]
…start by rewriting, then using implicit …
[pic]
[pic]
[pic]
[pic]
…now plug in the coordinates for x and y…to get slope…
[pic]
…write equation…[pic]
…or…[pic]
-----------------------
2
1
(
y
x
1
ADJ?
................
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