Download 1. Basic Derivative formulae (xn) = nx (ax) = a x ln a (ex) = e (loga x

1. Basic Derivative formulae
(xn )0 = nxn−1
(ax )0 = ax ln a
(loga x)0 =
(ex )0 = ex
1
x ln a
(ln x)0 =
1
x
(sin x)0 = cos x
(cos x)0 = − sin x
(tan x)0 = sec2 x
(cot x)0 = − csc2 x
(sec x)0 = sec x tan x
(csc x)0 = − csc x cot x
1
1 − x2
1
(tan−1 x)0 =
1 + x2
1
(sec−1 x)0 = √
x x2 − 1
(cos−1 x)0 = √
−1
1 − x2
−1
(cot−1 x)0 =
1 + x2
−1
−1
0
(csc x) = √
x x2 − 1
(sin−1 x)0 = √
2. Differentiation Rules
Sum rule: (f + g)0 = f 0 + g 0
wheref = f (x), g = g(x)
Product rule: (f · g)0 = f 0 g + g 0 f
0
f 0 g − g0 f
f
=
Quotient rule:
g
g2
0
Chain rule: [f (g)] = f 0 (g) · g 0
or
dy
=
dx
dy
du
du
·
dx
Implicit differentiation:
If y = y(x) is given implicitly, find derivative to the entire equation with respect to x. Then solve for y 0 .
3. Identities of Trigonometric Functions
sin x
cos x
1
sec x =
cos x
tan x =
sin2 x + cos2 x = 1
cos x
sin x
1
csc x =
sin x
cot x =
1 + tan2 x = sec2 x
1 + cot2 x = csc2 x
4. Laws of Exponential Functions and Logarithms Functions
ax · ay = ex+y
loga (xy) = loga (x) + loga (y)
ax
= ax−y
ay
loga ( xy ) = loga (x) − loga (y)
(ax )y = axy
loga (xn ) = n loga (x)
aloga (x) = x
ln x = loge x
a0 = 1
loga a = 1, loga 1 = 0
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