Benginning Calculus Lecture notes 6 - implicit differentiation

Beginning Calculus
- Implicit Di¤erentiation and Inverse Functions -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
(D5) Implicit Di¤erentiation and Inverse Functions 1 / 16

Implicit Di¤erentiation Derivative of Inverse Functions
Leraning Outcomes
Find the derivative of functions explicitly and Implicitly.
Compute the derivatives of Inverse Functions

Know:
d
dx
(xn
) = nxn 1
, n 2 Z
Don’t know:
d
dx
xm/n
,
m
n
2 Q, n 6= 0

Let y = xm/x .
y = xm/n
yn
= xm
d
dx
(yn
) =
d
dx
(xm
)
d
dy
(yn
)
dy
dx
= mxm 1
, chain rule:
dyn
dx
=
dyn
dy
dy
dx
nyn 1 dy
dx
= mxm 1
dy
dx
=
mxm 1
nyn 1
=
m
n
xm 1
xm/n n 1
= axa 1
with a =
m
n

Let x2 + y2 = 25. This is not a function. The equation implicitly de…nes
y as several functions of x.
-2 -1 1 2
-2
-1
1
2
x
y
x2
+ y2
= 1
y =
p
1 x2

Explicit Di¤erentiation
Take the derivatives of y =
p
1 x2 and y =
p
1 x2 :
d
dx
p
1 x2 =
d
dx
1 x2
1/2
=
1
2
1 x2
1/2
( 2x)
=
x
p
1 x2
d
dx
p
1 x2 =
d
dx
1 x2
1/2
=
1
2
1 x2
1/2
( 2x)
=
x
p
1 x2

But sometimes it is not easy to di¤erentiate such equations, for example
y4
+ xy2
= 2
) y2
=
x
p
x2 4 ( 2)
2
) y =
s
x
p
x2 + 8
2
for y de…ned explicitly as a function of x.

Example
d
dx
x2 + y2 = 1 :
dy
dx
=
x
y
=
x
p
1 x2

Example
d
dx
x3 + y3 = 6xy :
dy
dx
=
x2 2y
2x y2

Example
d
dx
y4 + xy2 = 2 :
d
dx
y4
+ xy2
=
d
dx
2
4y3 dy
dx
+ y2
+ 2xy
dy
dx
= 0
dy
dx
=
y2
4y3 + 2xy
At x = 1, y = 1. So,
d
dx
y4
+ xy2
= 2
x=1
=
1
6

Example
d
dx
sin (x + y) = y2 cos x :
dy
dx
=
cos (x + y) + y2 sin x
2y cos x cos (x + y)

Example
d2
dx2
x4 + y4 = 16 :
d2y
dx2
=
48x2
y7
First, …nd
dy
dx
. Then
d
dx
dy
dx
.

The Inverse Function
Let y =
p
x, x > 0. Then, y2 = x. If we let f (x) =
p
x and
g (y) = x, then g (y) = y2.
-4 -2 2 4
-4
-2
2
4 g
f

In general, If y = f (x) and g (y) = x, then g (f (x)) = x.
g = f 1 and f = g 1.
y = f (x) , f 1
(y) = x
f f 1
(x) = x
Implicit di¤erentiation allows us to …nd the derivative of any inverse
function provided we know the derivative of the function.

Example
Let y = sin 1 x.
d
dx
sin 1 x =
1
p
1 x2
Use: sin y = x.

Example
Let y = tan 1 x. (Note: tan 1 x = arctan x).
d
dx
tan 1 x = cos2 y =
1
x2 + 1
Use: tan y = x.

Benginning Calculus Lecture notes 6 - implicit differentiation

More Related Content

What's hot (20)

Viewers also liked (12)

Similar to Benginning Calculus Lecture notes 6 - implicit differentiation (9)

Recently uploaded (20)

Benginning Calculus Lecture notes 6 - implicit differentiation