Implicit Differentiation
Lesson 6.4
Tangent to a Circle
Consider the graph of the equation shown.
x 2  y 2  64
Try this on the
spreadsheet
How can we use
calculus to find the
slope of a tangent for
a particular (x, y) on
the circle?
Why is this a
problem?
2
Explicit Functions
We have worked with functions of the form
y  f ( x)
Examples:
y  4x  5
y  x  x6
2
x3
y 2
x
Even when given 2x – 3y = 12
• We can solve for
2
y  x4
3
3
Implicit Functions
Some functions cannot be readily solved
for y.
y  3xy  6 x  0
2
2
x2 y 2

1
25 121
For these we say y is given implicitly in
terms of x
dy
We will use implicit differentiation to find
dx
4
Implicit Differentiation
Given 4xy – 6y2 = 8
• We differentiate with respect to x on both sides
of the equation
d
d
2
4
xy

6
y

10 


dx
dx
Each time an
expression has a y
in it, we use the
chain rule
0
d
d
d
d d 2
 4 x  y   4 x  y  6 yy 4 x  12 y  dx ( y)
dx dx
dx
dx
dy
dy
Use product rule
4 y  4x
12 y
0
Use chain rule
and chain rule
dx
dx
5
Implicit Differentiation
dy
Now we have an equation and solve for
dx
dy
dy
4 y  4 x  12 y
0
dx
dx
dy
dy
4 y  12 y  4 x
dx
dx
dy
4 y  12 y  4 x 
dx
dy
4x

dx 12 y  4 x
6
We Better Try This Again
Find dy/dx for following
x  4 y  10
3
2
2 x  y 1
x e yx
2 y
3
7
Tangent Lines
Consider the equation for a circle
• x2 + y2 = 36
• What is the equation
of the tangent to the
circle at the point
where x = 5 in the
4th quadrant
Find the slope by using implicit
differentiation
• Substitute in (5, -3.316)
Use point-slope formula for line
8
Review
To find dy/dx for an equation containing both
x and y
1. Differentiate both sides of equation
w/respect to x
•
Place all terms with dy/dx on one side
2.
•
3.
4.
Assuming y is a function of x
All others on other side
Factor out dy/dx
Solve for dy/dx
9
Implicit Differentiation on the TI
Calculator
We can declare a function which will do
implicit differentiation:
Note, this cannot
be an equation,
only an
expression
Usage:
10
Assignment
Lesson 6.4
Page 401
Exercises 1 – 21 odd, 43
11