Math 1205 Calculus/Sec. 3.6 Implicit Differentiation
I.
Introduction
A. Explicit Functions
1. Defn: An explicit function is a function in which one variable is defined only in terms of
the another variable.
2. Examples
a. y = 3x + 7
slope intercept form of a line
y = 9− x
2
c. y = − 9 − x
d. y = cos θ
2
b.
top half of a circle centered at (0,0) with radius 3
bottom half of a circle centered at (0,0) with radius 3
B. Implicit Functions
1. Defn: An implicit function is a function in which one variable is not defined only in terms
of the another variable.
2. Examples
a. −3x +
y=7
b. x + y = 9
2
3 3
c. y + 4x y − 9x = sin xy
2
2
general form of a line
full circle centered at (0,0) with radius 3
3. Some implicit functions can be made into explicit functions
a. Both examples (a) from above represent the same line.
b. Example (b) above gives rise to both of the half circles depending on which range you
want.
c. Example (c) above cannot be solved for y in terms of x.
II.
Implicit Differentiation
A. Introduction
Previously all functions we have dealt with were of the form y=f(x) . We now want to look at
implicit equations where we can’t get y explicitly in terms of x or the resulting explicit
function is too complicated to deal with.
B. Steps to find
dy
of implicit functions involving the variables x and y using implicit
dx
differentiation
1. Treat
y as differentiable function of x.
2. Differentiate both sides of the equation wrt
3. Solve for
x using all the rules we have previously used.
dy
.
dx
4. Simplify each side of the equal sign.
dy
dy
terms to one side of the equal sign and all non
terms to the other.
dx
dx
dy
6. Factor out
.
dx
dy
7. Divide to isolate
.
dx
5. Bring all
C. Examples:
1.
Differentiate the following
−3x + y = 7 ;
dy
dx
compared with
dy
dx
2.
x 5 + 4y 3 = 7 − 3y 5 ;
3.
2xy 3 + y 2 = cos(x + y) ;
dy
dx
y = 3x + 7
4. Find the tangent line to the curve
5. Find the normal line to the curve
III.
x 2 + y 2 = 25 at the pt (3,-4).
6x 2 + 3xy + 2y 2 + 17y = 6 at the pt (-1,0).
Derivatives of Higher Order Using Implicit Differentiation
A. Steps to find
d2y
of implicit functions involving the variables x and y using implicit
dx 2
differentiation
1. Find
dy
implicitly.
dx
d2y
2. Differentiate both sides of the equation wrt x. The left hand side of the equation is
dx 2
dy
and the right hand side will include x, y and
terms. (Remember, every time you
dx
dy
implicitly differentiate the variable y, you must include
).
dx
3. Substitute in the value of
dy
dy
d2y
found in step 1 in for all
to express
in terms of x
dx
dx
dx 2
and y.
4. Simplify the right hand side.
5. Simplify the right hand side further using the original function.
B. Example
dy
d 2y
1. Find
and
for
dx
dx 2
2.
Find
x + y =1.
dy
d 2y
2
and
2 for xy + y = 1
dx
dx
IV.
Additional Examples
A. Find
ds
2
3
5
for s = 4st + 3t − 8
dt
dr
r3
2
4
B. Find
for θ + 3r + sin(rθ ) =
dθ
θ