Page 79 - Schaum's Outline of Theory and Problems of Advanced Calculus

70 DERIVATIVES [CHAP. 4

RULES FOR DIFFERENTIATION
If f , g; and h are differentiable functions, the following differentiation rules are valid.
d d d
1: 0 0 (Addition Rule)
dx f f ðxÞþ gðxÞg ¼ dx f ðxÞþ dx gðxÞ¼ f ðxÞþ g ðxÞ

d d d
2: 0 0
dx f f ðxÞ gðxÞg ¼ dx f ðxÞ dx gðxÞ¼ f ðxÞ g ðxÞ

d d
3: fCf ðxÞg ¼ C f ðxÞ¼ Cf ðxÞ where C is any constant
0
dx dx
d d d
4: f f ðxÞgðxÞg ¼ f ðxÞ gðxÞþ gðxÞ f ðxÞ¼ f ðxÞg ðxÞþ gðxÞ f ðxÞ (Product Rule)
0
0
dx dx dx
d d
d gðxÞ dx f ðxÞ f ðxÞ dx gðxÞ 0 0
5: f ðxÞ gðxÞ f ðxÞ f ðxÞg ðxÞ if gðxÞ 6¼ 0 (Quotient Rule)
2 ¼ 2
¼
dx gðxÞ
½gðxފ ½gðxފ
6: If y ¼ f ðuÞ where u ¼ gðxÞ; then

dy dy du du
0 0 0
dx ¼ du dx ¼ f ðuÞ dx ¼ f fgðxÞgg ðxÞ ð12Þ

Similarly if y ¼ f ðuÞ where u ¼ gðvÞ and v ¼ hðxÞ, then

dy dy du dv
dx ¼ du dv dx ð13Þ

The results (12) and (13) are often called chain rules for differentiation of composite functions.

7: If y ¼ f ðxÞ; and x ¼ f 1 ðyÞ; then dy=dx and dx=dy are related by

dy 1
dx ¼ dx=dy ð14Þ

8: If x ¼ f ðtÞ and y ¼ gðtÞ; then

dy dy=dt g ðtÞ
0
dx ¼ dx=dt ¼ f ðtÞ ð15Þ
0
Similar rules can be formulated for differentials. For example,

df f ðxÞþ gðxÞg ¼ df ðxÞþ dgðxÞ¼ f ðxÞdx þ g ðxÞdx ¼f f ðxÞþ g ðxÞgdx
0
0
0
0
df f ðxÞgðxÞg ¼ f ðxÞdgðxÞþ gðxÞdf ðxÞ¼f f ðxÞg ðxÞþ gðxÞ f ðxÞgdx
0
0