Contributed by:
In this section, we will learn: How functions are defined implicitly.
1.
3
DIFFERENTIATION RULES
2.
DIFFERENTIATION RULES
The functions that we have met so far can be
described by expressing one variable
explicitly in terms of another variable.
For example, y = x 3 + 1 , or y = x sin x,
or in general y = f(x).
3.
DIFFERENTIATION RULES
However, some functions are
defined implicitly.
4.
DIFFERENTIATION RULES
3.5
Implicit Differentiation
In this section, we will learn:
How functions are defined implicitly.
5.
IMPLICIT DIFFERENTIATION Equations 1 and 2
Some examples of implicit functions
x2 + y2 = 25
x3 + y3 = 6xy
6.
IMPLICIT DIFFERENTIATION
In some cases, it is possible to solve such an
equation for y as an explicit function (or
several functions) of x.
For instance, if we solve Equation 1 for y,
we get y = ± 25 −x 2
So, two of the functions determined by
the implicit Equation 1 are f ( x) = 25 −x 2
and g ( x) = − 25 −x 2
7.
IMPLICIT DIFFERENTIATION
The graphs of f and g are the upper
and lower semicircles of the circle
x2 + y2 = 25.
8.
IMPLICIT DIFFERENTIATION
It’s not easy to solve Equation 2 for y
explicitly as a function of x by hand.
A computer algebra system has no trouble.
However, the expressions it obtains are
very complicated.
9.
FOLIUM OF DESCARTES
Nonetheless, Equation 2 is the equation
of a curve called the folium of Descartes
shown here and it implicitly defines y as
several functions of x.
10.
FOLIUM OF DESCARTES
The graphs of three functions defined by
the folium of Descartes are shown.
11.
IMPLICIT DIFFERENTIATION
When we say that f is a function defined
implicitly by Equation 2, we mean that
the equation x3 + [f(x)]3 = 6x f(x) is true for
all values of x in the domain of f.
12.
IMPLICIT DIFFERENTIATION
Fortunately, we don’t need to solve
an equation for y in terms of x to find
the derivative of y.
13.
IMPLICIT DIFFERENTIATION METHOD
Instead, we can use the method of
implicit differentiation.
This consists of differentiating both sides of
the equation with respect to x and then solving
the resulting equation for y’.
14.
IMPLICIT DIFFERENTIATION METHOD
In the examples, it is always assumed that
the given equation determines y implicitly as
a differentiable function of x so that the
method of implicit differentiation can be
15.
IMPLICIT DIFFERENTIATION Example 1
dy
a. If x2 + y2 = 25, find .
dx
b. Find an equation of the tangent to
the circle x2 + y2 = 25 at the point (3, 4).
16.
IMPLICIT DIFFERENTIATION Example 1 a
Differentiate both sides of the equation
x2 + y2 = 25:
d 2 2 d
( x + y ) = (25)
dx dx
d 2 d 2
(x ) + ( y ) = 0
dx dx
17.
IMPLICIT DIFFERENTIATION Example 1 a
Remembering that y is a function of x and
using the Chain Rule, we have:
d 2 d 2 dy dy
(y ) = (y ) = 2y
dx dy dx dx
dy
2x + 2 y =0
dx
dy dy x
Then, we solve this equation for : =−
dx dx y
18.
IMPLICIT DIFFERENTIATION E. g. 1 b—Solution 1
At the point (3, 4) we have x = 3 and y = 4.
So, dy = −3
dx 4
Thus, an equation of the tangent to the circle at (3, 4)
is: y – 4 = – ¾(x – 3) or 3x + 4y = 25.
19.
IMPLICIT DIFFERENTIATION E. g. 1 b—Solution 2
Solving the equation x2 + y2 = 25,
2
we get: y = ± 25 −x
The point (3, 4) lies on the upper semicircle y = 25 −x 2
So, we consider the function f ( x) = 25 −x 2
20.
IMPLICIT DIFFERENTIATION E. g. 1 b—Solution 2
Differentiating f using the Chain Rule,
we have:
2 −1/ 2d 2
f '( x) = (25 −x )
1
2 (25 −x )
dx
2 −1/ 2
= 2 (25 −x ) (−2 x)
1
x
=−
2
25 −x
21.
IMPLICIT DIFFERENTIATION E. g. 1 b—Solution 2
So, 3
3
f '(3) = − =−
25 −32 4
As in Solution 1, an equation of the tangent is
3x + 4y = 25.
22.
NOTE 1
The expression dy/dx = -x/y in Solution 1
gives the derivative in terms of both x and y.
It is correct no matter which function y is
determined by the given equation.
23.
NOTE 1
2
For instance, for y = f ( x) = 25 −x ,
we have: dy x x
=− =−
dx y 25 −x 2
However, for y = g ( x) = − 25 −x 2 ,
we have:
dy x x x
=− =− =
dx y − 25 −x 2
25 −x 2
24.
IMPLICIT DIFFERENTIATION Example 2
a. Find y’ if x3 + y3 = 6xy.
b. Find the tangent to the folium of Descartes
x3 + y3 = 6xy at the point (3, 3).
c. At what points in the first quadrant is
the tangent line horizontal?
25.
IMPLICIT DIFFERENTIATION Example 2 a
Differentiating both sides of x3 + y3 = 6xy
with respect to x, regarding y as a function
of x, and using the Chain Rule on y3 and
the Product Rule on 6xy, we get:
3x2 + 3y2y’ = 6xy’ + 6y
or x2 + y2y’ = 2xy’ + 2y
26.
IMPLICIT DIFFERENTIATION Example 2 a
Now, we solve for y’:
2 2
y y '−2 xy ' = 2 y −x
2 2
( y −2 x) y ' = 2 y −x
2
2 y −x
y' = 2
y −2 x
27.
IMPLICIT DIFFERENTIATION Example 2 b
2
When x = y = 3, 2 ⋅3 −3
y'= 2 = −1
3 −2 ⋅3
A glance at the figure confirms
that this is a reasonable value
for the slope at (3, 3).
So, an equation of the tangent
to the folium at (3, 3) is:
y – 3 = – 1(x – 3) or x + y = 6.
28.
IMPLICIT DIFFERENTIATION Example 2 c
The tangent line is horizontal if y’ = 0.
Using the expression for y’ from (a), we see that y’ = 0
when 2y – x2 = 0 (provided that y2 – 2x ≠ 0).
Substituting y = ½x2 in the equation of the curve,
we get x3 + (½x2)3 = 6x(½x2) which simplifies to
x6 = 16x3.
29.
IMPLICIT DIFFERENTIATION Example 2 c
Since x ≠ 0 in the first quadrant,
we have x3 = 16.
If x = 161/3 = 24/3, then y = ½(28/3) = 25/3.
30.
IMPLICIT DIFFERENTIATION Example 2 c
Thus, the tangent is horizontal at (0, 0)
and at (24/3, 25/3), which is approximately
(2.5198, 3.1748).
Looking at the figure, we see
that our answer is reasonable.
31.
NOTE 2
There is a formula for the three roots
of a cubic equation that is like
the quadratic formula, but much more
32.
NOTE 2
If we use this formula (or a computer algebra
system) to solve the equation x3 + y3 = 6xy
for y in terms of x, we get three functions
determined by the following equation.
33.
NOTE 2
y = f ( x) = 3 −12 x 3 + 1
4 x 6 −8 x 3 + 3 −12 x 3 − 1
4 x 6 −8 x3
⎡ ⎤
y = ⎢−f ( x) ± −3 ⎛
1
2
3
−1 3
⎜ 2x +
1
4
6 3
x −8 x − − x −
3 1
2
3 1
4
6 3⎞
x −8 x ⎟⎥
⎣ ⎝ ⎠⎦
34.
NOTE 2
These are the three functions whose
graphs are shown in the earlier figure.
35.
NOTE 2
You can see that the method of implicit
differentiation saves an enormous amount of
work in cases such as this.
36.
NOTE 2
Moreover, implicit differentiation works just
as easily for equations such as
y5 + 3x2y2 + 5x4 = 12
for which it is impossible to find a similar
expression for y in terms of x.
37.
IMPLICIT DIFFERENTIATION Example 3
Find y’ if sin(x + y) = y2 cos x.
Differentiating implicitly with respect to x and
remembering that y is a function of x, we get:
cos( x + y ) ⋅(1 + y ') = y 2 (−sin x) + (cos x)(2 yy ')
Note that we have used the Chain Rule on the left side
and the Product Rule and Chain Rule on the right side.
38.
IMPLICIT DIFFERENTIATION Example 3
If we collect the terms that involve y’,
we get:
cos( x + y ) + y 2 sin x = (2 y cos x) y '−cos( x + y ) ⋅y '
y 2 sin x + cos( x + y )
So, y ' =
2 y cos x −cos( x + y )
39.
IMPLICIT DIFFERENTIATION Example 3
The figure, drawn with the implicit-plotting
command of a computer algebra system,
shows part of the curve sin(x + y) = y2 cos x.
As a check on our calculation,
notice that y’ = -1 when
x = y = 0 and it appears that
the slope is approximately -1
at the origin.
40.
IMPLICIT DIFFERENTIATION
The following example shows how to
find the second derivative of a function
that is defined implicitly.
41.
IMPLICIT DIFFERENTIATION Example 4
Find y” if x4 + y4 = 16.
Differentiating the equation implicitly with
respect to x, we get 4x3 + 4y3y’ = 0.
42.
IMPLICIT DIFFERENTIATION E. g. 4—Equation 3
Solving for y’ gives:
3
x
y' = − 3
y
43.
IMPLICIT DIFFERENTIATION Example 4
To find y’’, we differentiate this expression
for y’ using the Quotient Rule and
remembering that y is a function of x:
d ⎛ x3 ⎞ y 3 (d / dx)( x 3 ) −x 3 (d / dx)( y 3 )
y '' = ⎜− 3 ⎟ = − 3 2
dx ⎝ y ⎠ (y )
y 3 ⋅3 x 2 −x 3 (3 y 2 y ')
=− 6
y
44.
IMPLICIT DIFFERENTIATION Example 4
If we now substitute Equation 3 into
this expression, we get:
2 3 3 2 ⎛ x ⎞
3
3 x y −3 x y ⎜− 3 ⎟
⎝ y ⎠
y '' = − 6
y
2 4 6 2 4 4
3( x y + x ) 3x ( y + x )
=− 7
=− 7
y y
45.
IMPLICIT DIFFERENTIATION Example 4
However, the values of x and y must satisfy
the original equation x4 + y4 = 16.
So, the answer simplifies to:
2 2
3 x (16) x
y '' = − 7
= −48 7
y y
46.
INVERSE TRIGONOMETRIC FUNCTIONS (ITFs)
The inverse trigonometric functions were
reviewed in Section 1.6
We discussed their continuity in Section 2.5 and
their asymptotes in Section 2.6
47.
DERIVATIVES OF ITFs
Here, we use implicit differentiation to find
the derivatives of the inverse trigonometric
functions—assuming that these functions are
48.
DERIVATIVES OF ITFs
In fact, if f is any one-to-one differentiable
function, it can be proved that its inverse
function f -1 is also differentiable—except
where its tangents are vertical.
This is plausible because the graph of a differentiable
function has no corner or kink.
So, if we reflect it about y = x, the graph of its inverse
function also has no corner or kink.
49.
DERIVATIVE OF ARCSINE FUNCTION
Recall the definition of the arcsine function:
−1 π π
y = sin x means sin y = x and − ≤ y ≤
2 2
Differentiating sin y = x implicitly with respect
to x, we obtain:
dy dy 1
cos y = 1 or =
dx dx cos y
50.
DERIVATIVE OF ARCSINE FUNCTION
Now, cos y ≥ 0, since –π/2 ≤ y ≤ π/2.
So, cos y = 1 −sin 2 y = 1 −x 2
Thus, dy = 1 = 1
dx cos y 1 −x 2
d −1 1
(sin x) =
dx 1 −x 2
51.
DERIVATIVE OF ARCTANGENT FUNCTION
The formula for the derivative of the
arctangent function is derived in a similar way.
If y = tan -1x, then tan y = x.
Differentiating this latter equation implicitly
with respect to x, we have:
2 dy
sec y =1
dx
dy 1 1 1
= = =
dx sec y 1 + tan y 1 + x 2
2 2
d −1 1
(tan x) =
dx 1 + x2
52.
DERIVATIVES OF ITFs Example 5
1
a. y = −1
sin x
b. f(x) = x arctan x
53.
DERIVATIVES OF ITFs Example 5 a
dy d −1 −1 −1 −2 d −1
= (sin x) = −(sin x) (sin x)
dx dx dx
1
=−
−1 2 2
(sin x) 1 −x
54.
DERIVATIVES OF ITFs Example 5 b
1 −1/ 2
f '( x) = x 2
(
1
2 x ) + arctan x
1+ ( x )
x
= + arctan x
2(1 + x)
55.
DERIVATIVES OF ITFs
The inverse trigonometric functions
that occur most frequently are the ones
that we have just discussed.
56.
DERIVATIVES OF ITFs
The derivatives of the remaining four are
given in this table.
The proofs of the formulas are left as exercises.
d −1 1 d −1 1
(sin x) = (csc x) = −
dx 1 −x 2 dx x x 2 −1
d −1 1 d −1 1
(cos x) = − (sec x) =
dx 1 −x 2 dx x x 2 −1
d −1 1 d −1 1
(tan x) = (cot x) = −
dx 1 + x2 dx 1 + x2
+1-315-636-4485
+91 98151-74050 
