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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