Page 252 - DMTH201_Basic Mathematics-1

Unit 9: Differential Calculus

This can be remembered as Notes
d d
Dr Nr Nr Dr
d Nr dx dx
dx Dr Dr 2

where Nr Numerator, Dr Denominator

Notes (1) While doing problems on differentiation, the above theorems should be
strictly followed.

(2) The above theorems can be proved using the method of first principles. Since
there is no mention of proofs of the theorems in the syllabus, the proofs are
not given here. Only Statements are given because they have to be used in the
problems.

9.3 Derivatives of Standard Functions

d n n 1
(1) x nx
dx
d
(2) x 1
dx
d 1 1
(3) 2
dx x x
d 1
(4) x
dx 2 x
d x x
(5) e e
dx
d x x
(6) a a log a where a 0 and a 1
e
dx
d 1
(7) log x where x 0
e
dx x
d 1
(8) log x
a
dx x log a
e
d n n 1
(9) ax b na ax b
dx
d 1 1 1
(10) 2
dx ax b a ax b
d ax b ax b
(11) e ae
dx
d 1
(12) log ax b a
e
dx ax b

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