Derivatives Flashcards
1
Q
Limit definition of a derivative
A
f’(x) = lim(h->0) (f(x + h) - f(x))/h if the limit exists
2
Q
m = f’(a) is the slope of?
A
The tangent line to y = f(x) at x = a
3
Q
Equation of the tangent line
A
y = f(a) + f’(a)(x-a)
4
Q
The tangent line can be?
A
Instantaneous rate of change at a point, velocity
5
Q
a’
A
0
6
Q
[f(x) + g(x)]’
A
f’(x) + g’(x)
7
Q
[f(x)g(x)]’
A
f(x)g’(x) + f’(x)g(x)
8
Q
[f(g(x)]’
A
f’(g(x))g’(x)
9
Q
[cf(x)]’
A
cf’(x)
10
Q
[f(x) - g(x)]’
A
f’(x) - g’(x)
11
Q
[f(x)/g(x)]’
A
(f’(x)g(x) - f(x)g’(x))/(g(x))^2
12
Q
[f^-1(x)]’
A
1/(f’(f^-1(x)))
13
Q
[x^a]’
A
ax^(a-1)
14
Q
[e^x]’
A
e^x
15
Q
[e^u]’
A
u’e^u
16
Q
[a^x]’
A
a^(x)lna
17
Q
[a^u]’
A
a^(u)u’lna
18
Q
[lnx]’
A
1/x
19
Q
[lnu]’
A
u’/u
20
Q
[log(a)x]’
A
1/xlna
21
Q
[log(a)u]’
A
u’/ulna
22
Q
[sinx]’
A
cosx
23
Q
[cosx]’
A
-sinx
24
Q
[tanx]’
A
sec^(2)x
25
[cotx]'
-csc^(2)x
26
[secx]'
secxtanx
27
[cscx]'
-cscxcotx
28
[sinu]'
u'cosu
29
[cosu]'
-u'sinu
30
[tanu]'
u'sec^(2)u
31
[cotu]'
-u'csc^(2)u
32
[secu]'
u'secutanu
33
[cscu]'
-u'cscucotu
34
[arcsinx]'
1/sqrt(1-x^2)
35
[arccosx]'
-1/sqrt(1-x^2)
36
[arctanx]'
1/(1+x^2)
37
[arccotx]'
-1/(1+x^2)
38
[arcsecx]'
1/|x|sqrt(x^(2) - 1)
39
[arccscx]'
-1/|x|sqrt(x^(2) - 1)
40
[arcsinu]'
u'/sqrt(1-u^2)
41
[arccosu]'
-u'/sqrt(1-u^2)
42
[arctanu]'
u'/(1 + u^2)
43
[arccotu]'
-u'/(1 + u^2)
44
[arcsecu]'
u'/|u|sqrt(u^(2) - 1)
45
[arccscu]'
-u'/|u|sqrt(u^(2) - 1)
