Page 45 - Aeronautical Engineer Data Book

32 Aeronautical Engineer’s Data Book

d d u d 1 n du
n
3
3
3 3 (u ) = nu n–1 3 3,
3 3 �
�
3 3 3
3 = –
d x dx d x u n u n+1 d x
d u dx dx
3 3 = 1 3 , 3 if 3 ≠ 0
3
dx / u d u
d
d d u
3 3 f (u) = f’(u) 3
3
d x dx
d x
3 3 � f(t)dt = f(x)
d x a
d b
3 3 � f(t)dt = – f(x)
d x x
b
d � f(x, t)dt = � b ∂ f
3 3 33 dt
d x a a ∂x
v
d � f(x, t)dt = � u ∂ f d v
3 3 33 dt + f (x, v) 3 3
d x u v ∂ x d x
d u
– f (x, u) 3 3
d x
Higher derivatives
�
2
d d y d y
Second derivatives = 33 � 3 3 = 3 3
d x d x dx 2
= f"(x) = y"
2
d u
d 2 � � 2 d u
33 f(u) = f "(u) 3 3 + f'(u) 3 3
2
2
d x d x dx
Derivatives of exponentials and logarithms
d
n
3 3 (ax + b) = na(ax + b) n–1
d x
d
3 3 e = ae ax
ax
d x
d 1
3
3 3 ln ax = 3, ax > 0
d x x