Page 264 - DMTH201_Basic Mathematics-1

Unit 9: Differential Calculus

1 2 Notes
2 3x 3 3
3
1
2
3 2 3x

11. log x 1 x 2

Solution: Let y log x 1 x 2

dy 1 d 2
. x 1 x
dx x 1 x 2 dx

1 1 d 2
. 1 . 1 x
x 1 x 2 2 1 x 2 dx

1 1
. 1 . 2x
x 1 x 2 2 1 x 2
1 x
. 1
x 1 x 2 1 x 2

1 1 x 2 x
.
x 1 x 2 1 x 2

x 1 x 2
x 1 x 2 1 x 2

1
1 x 2

Task Differentiate following functions w.r.t. x
(1) log(log(log ))
x
1 x 2
(2) log
1– x 2

Example
A spherical balloon is being inflated at the rate of 5 cubic feet per second. How fast is the radius
of the balloon increasing 15 seconds after the start?

Solution:
Let v denote volume and r denote radius of the balloon. Since v is changing with time, r will also
4
change. Volume of a spherical balloon v r 3 . Here v is a function of r and r is a function of
3

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