1. Aim: What Is Implicit Differentiation and How Does It Work?
Do Now:
Explicit

2. Implicit vs. Explicit
Explicit Form
Implicit Form
variable y is written as a function of x
derivative of y?
Often you can solve for y in term of x
Not Always!
Implicit Differentiation is used

3. Differentiating with Respect to x
Use Simple Power Rule
variables agree un
nun-1 u’
Use Chain Rule
variables disagree
Chain Rule
Product Rule
Chain Rule
Simplify

4. Differentiating with Respect to x
Use Simple Power Rule
variables agree un
nun-1 u’
COMMON ERROR! DON’T FORGET
Use Chain Rule
variables disagree
Chain Rule
Product Rule
Chain Rule
Simplify

5. Guidelines for Implicit Differentiation
Differentiate both sides of the equation with respect to x.
Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.
Factor dy/dx out of the left side of the equation.
Solve for dy/dx by dividing both sides of the equation by the left-hand factor that does not contain dy/dx.

6. Find dy/dx given y3 + y2 – 5y – x2 = -4
Model Problem
Find dy/dx given y3 + y2 – 5y – x2 = -4
Differentiate both sides of the equation with respect to x.
2. Collect all terms involving dy/dx on the left side of the equation

7. Find dy/dx given y3 + y2 – 5y – x2 = -4
Model Problem
Find dy/dx given y3 + y2 – 5y – x2 = -4
3. Factor dy/dx out of the left side of the equation.
4. Solve for dy/dx by dividing by (3y2 + 2y – 5)
y3 + y2 – 5y – x2 = -4
function? NO
(1, 1)
(2, 0)
(1, -3)
slope at (1, 1)
slope at (2, 0)
slope at (1, -3)
und
-4/5
1/8

8. Functions from Equations
If a segment of a graph can be represented by a differentiable function, dy/dx will have meaning as the slope.
function? NO
YES
YES
Recall: a function is not differentiable at points with vertical tangents
nor at points where the function is not continuous

9. Aim: What Is Implicit Differentiation and How Does It Work?
Do Now:
Determine the slope of the tangent line to the graph x2 + 4y2 = 4 at the point .

10. Model Problem
Determine the slope of the tangent line to the graph x2 + 4y2 = 4 at the point .
implicit differentiation
solve for dy/dx
evaluate for the point
Slope of tangent at is 1/2

11. Model Problem
Determine the slope of the tangent line to the graph 3(x2 + y2)2 = 100xy and the point (3, 1).
Constant and General Power Rules
FOIL and isolate dy/dx

12. Model Problem
Determine the slope of the tangent line to the graph 3(x2 + y2)2 = 100xy and the point (3, 1).

13. Model Problem
Determine the slope of the tangent line to the graph 3(x2 + y2)2 = 100xy and the point (3, 1).
substitute (3, 1)

14. Finding the 2nd Derivative Implicitly
Given x2 + y2 = 25, find
find first derivative implicitly:
quotient rule
sub –x/y for dy/dx
sub 25 for x2+y2

15. Model Problem
Find the tangent line to the graph given by x2(x2 + y2) = y2 at the point
implicit differentiation

16. point-slope formula for equation (y – y1) = m(x – x1)
Model Problem
Find the tangent line to the graph given by x2(x2 + y2) = y2 at the point
substitute
= 3 = m
point-slope formula for equation (y – y1) = m(x – x1)

17. Model Problem
Find dy/dx implicitly for the equation sin y = x. Then find the largest interval of the form –a < y < a such that y is a differentiable function of x.
explicitly:
for the interval -/2 < y < /2, we use and substitute the original equation to arrive at