This document discusses implicit differentiation and derivatives of inverse functions. It begins by introducing implicit differentiation and examples of taking derivatives of functions defined implicitly rather than explicitly. It then covers using implicit differentiation to find derivatives of inverse functions. Examples are provided such as finding the derivative of sin^-1(x) and tan^-1(x). The key concepts are using implicit differentiation to take derivatives where the relationship between variables is given implicitly rather than explicitly, and to take derivatives of inverse functions.

1. 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
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2. Implicit Di¤erentiation Derivative of Inverse Functions
Leraning Outcomes
Find the derivative of functions explicitly and Implicitly.
Compute the derivatives of Inverse Functions
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3. Implicit Di¤erentiation Derivative 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
VillaRINO DoMath, FSMT-UPSI
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4. Implicit Di¤erentiation Derivative of Inverse Functions
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
VillaRINO DoMath, FSMT-UPSI
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5. Implicit Di¤erentiation Derivative of Inverse Functions
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
VillaRINO DoMath, FSMT-UPSI
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6. Implicit Di¤erentiation Derivative of Inverse Functions
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
VillaRINO DoMath, FSMT-UPSI
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7. Implicit Di¤erentiation Derivative of Inverse Functions
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.
VillaRINO DoMath, FSMT-UPSI
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8. Implicit Di¤erentiation Derivative of Inverse Functions
Example
d
dx
x2 + y2 = 1 :
dy
dx
=
x
y
=
x
p
1 x2
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9. Implicit Di¤erentiation Derivative of Inverse Functions
Example
d
dx
x3 + y3 = 6xy :
dy
dx
=
x2 2y
2x y2
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10. Implicit Di¤erentiation Derivative of Inverse Functions
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
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11. Implicit Di¤erentiation Derivative of Inverse Functions
Example
d
dx
sin (x + y) = y2 cos x :
dy
dx
=
cos (x + y) + y2 sin x
2y cos x cos (x + y)
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12. Implicit Di¤erentiation Derivative of Inverse Functions
Example
d2
dx2
x4 + y4 = 16 :
d2y
dx2
=
48x2
y7
First, …nd
dy
dx
. Then
d
dx
dy
dx
.
VillaRINO DoMath, FSMT-UPSI
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13. Implicit Di¤erentiation Derivative of Inverse Functions
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
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14. Implicit Di¤erentiation Derivative of Inverse Functions
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.
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15. Implicit Di¤erentiation Derivative of Inverse Functions
Example
Let y = sin 1 x.
d
dx
sin 1 x =
1
p
1 x2
Use: sin y = x.
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16. Implicit Di¤erentiation Derivative of Inverse Functions
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.
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