1. Section 2.5 – Implicit Differentiation

2. Explicit Equations
The functions that we have differentiated and handled so far can be described by expressing one variable explicitly in terms of another variable. For example: Or, in general, y = f(x).

3. Can we take the derivative of these functions?
Implicit Equations
Some functions, however, are defined implicitly ( not in the form y = f(x) ) by a relation between x and y such as:
Can we take the derivative of these functions?
It is possible to solve some Implicit Equations for y, then differentiate:
Yet, it is difficult to rewrite most Implicit Equations explicitly. Thus, we must be introduced to a new technique to differentiate these implicit functions.

4. White Board Challenge
Solve for y:

5. *Reminder*
Technically the Chain Rule can be applied to every derivative:

6. Derivatives Involving the Dependent Variable (y)
Find the derivative of each expression
The Chain Rule is Required. a. b.
The derivative of y with respect to x is…
the derivative of y.
This is another way to write y prime.

7. Instructions for Implicit Differentiation
If y is an equation defined implicitly as a differentiable function of x, to find the derivative:
Differentiate both sides of the equation with respect to x. (Remember that y is really a function of x for part of the curve and use the Chain Rule when differentiating terms containing y)
Collect all terms involving dy/dx on the left side of the equation, and move the other terms to the right side.
Factor dy/dx out of the left side
Solve for dy/dx

8. Example 1 If is a differentiable function of x such that find .
Differentiate both sides.
Product AND Constant Multiple Rules
Chain Rule
Solve for dy/dx

9. Example 2 Find if . Differentiate both sides Product Rule
Chain Rule Twice

10. Example 3
Find if
Find the first derivative by Differentiating both sides.
Quotient Rule
Chain Rule
Remember:
Remember:
Now Find the Second Derivative

11. Example 4
Find the slope of a line tangent to the circle at the point
Find the derivative by differentiating both sides.
Chain Rule
Evaluate the derivative at x=5 and y=4.

12. White Board Challenge
Find the derivative of:

13. Find the derivative by differentiating both sides.
Example 5
If and , find
Find the derivative by differentiating both sides.
Chain Rule

14. Evaluate the derivative with the given information.
Example 5 (continued)
If and , find
Evaluate the derivative with the given information.

15. Example 6 Find an equation of the tangent to the circle at the point .
Now evaluate the derivative at x=3 and y=4.
Find the derivative by differentiating both sides.
Chain Rule
Use the Point-Slope Formula to find the equation of the tangent line

16. White Board Challenge
Find the second derivative of: