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]

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2

1

(

y

x

1

ADJ?

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