IMPLICIT DIFFERENTIATION

Implicit vs. Explicit Functions

Example of explicit form:

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, etc.

Example of two other equations in implicit form:

Implicit Form Explicit Form Derivative

??? ????

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)… because the dx matches the x3

Thinking about it from the chain rule point of view:

b)… because the dx does not match the y3

Thinking about it from the chain rule point of view:

c) … sum/difference rule, then implicit differentiation (think chain rule)

d) … use implicit differentiation with the product rule…

General Guidelines for Implicit Differentiation

  1. Differentiate both sides of an equation with respect to x.
  2. Collect terms on the left side, and move the others to the right side.
  3. Factor out the from the left side terms.
  4. Solve for, by dividing both sides by the remaining terms that was factored out of.

EXAMPLE 2:

Find for

EXAMPLE 3:

Rewrite each as a differentiable function:

a)

This is the equation of a single point therefore is not differentiable at all … so drop it right here!

b)

This is a circle with radius = 1. Isolate y …

…

… You can now use “normal” differentiation …

Compare to the use of implicit differentiation.

… …

c)

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 …

…

…now repeat with implicit differentiation, and compare…

…

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:

…

…Use implicit …

Solve for … … at …

… …

… Equation:

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):

…use implicit, chain and product rules

… distribute …

… Isolate terms … and factor …

… Solve for …

… plug in (3, 1) …

EXAMPLE 6:

Find for:

… Use implicit differentiation …

… Solve for …

…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? =

EXAMPLE 7:

Find for ***

… that is, find the second derivative …

… so use “implicit” to find first …

…

… now use the quotient rule … and replacewith

…simplify by replacing ***

EXAMPLE 8

Find the tangent line to the graph given by the equation:

at

…start by rewriting, then using implicit …

…now plug in the coordinates for x and y…to get slope…

…write equation…

…or…

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