Simple Rules for Differentiation



Implicit Differentiation

Objectives:

Students will be able to

• Calculate derivative of function defined implicitly.

• Determine the slope of the tangent line to a function defined implicitly at a specified point.

• Calculate the equation of the tangent line to a function defined implicitly at a specified point.

All of our functions so far have explicitly define y as a function of x. Not all functions can be easily defined in this way. Instead we can have functions that implicitly define y as a function of x.

Such implicitly defined functions can also be differentiated. The process will involve the use of the chain rule. The thing to keep in mind is that y is a function of x and thus can be written as[pic]. The derivative of [pic] is [pic]. Since [pic] is y, this can be rewritten as [pic] or [pic]. This will be important in our process of implicit differentiation.

Example 1:

Find [pic] for [pic].

Example 2:

Find [pic] for [pic].

Example 3:

Find [pic] for [pic].

Example 4:

Find [pic] for [pic].

Example 5:

Find the equation of the tangent line to the curve [pic]at the point (1, 1)

Example 6:

Find the equation of the tangent line to the curve [pic] at [pic]

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