Page 254 - DMTH201_Basic Mathematics-1

Unit 9: Differential Calculus

Notes
Example: Solve log x
10
d d log x
Solution: log x e (using change of base in logarithms)
10
dx dx log 10
e
1 d
log x
e
log 10 dx
e
1 1 1
log 10 x x log 10
e
e
Example: Solve xe x
d x d x x d
Solution: xe x e e x (product rule)
dx dx dx
xe x e x (1)

(x 1)e x

Example: Solve x log x
2
d 2 2 d d 2
Solution: x log x x log x log x x (product rule)
dx dx dx
2 1
x log x 2x
x
x
x 2 log x
x 1
Example: Solve
x – 1
d d
x 1 (x 1) x 1 (x 1)
d x 1 dx dx
Solution: 2 (Quotient rule)
dx x 1 x 1

x 1 1 0 x 1 1 0
(x 1) 2
x 1 x 1
2
x 1
2
x 1 2

2x 3
Task Difference
x – 7

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