Derivatives Of Unit circle Flashcards
1
Q
Sec(x)
A
Tan(x)Sec(x)
2
Q
Tan(x)
A
Sec^2(x)
3
Q
Cos(x)
A
-sin(x)
4
Q
Sin(x)
A
Cos(x)
5
Q
Csc(x)
A
-csc(x)cot(x)
Negative csc cot
6
Q
Ln(x)
Aka the ln(any #)
A
1/x
1/the #
7
Q
A^x
A
A^xln(a)
8
Q
F(g(x)
A
F’(g(x)g’(x)
Derivative of the outer times, derivative of the inner
9
Q
F/g
A
(gf’-fg’)/g^2
10
Q
Ln(e)
A
1
11
Q
Y’=dx/sqrt(1-x^2)
A
Arcsin(x)
12
Q
Y’ = dx/x(sqrt(x^2-1)
A
Arcsec(x)
13
Q
Y’ = dx/1+x^2
A
Arctan(x)
14
Q
1+cot^2(x)
A
Csc^2(x)
15
Q
Tan^2+1
A
Sec^2(x)
16
Q
1/sin(x)
A
Csc(x)
17
Q
Sum of sines
Sin (a+b)
A
Sin(a)Cos(b)cos(a)sin(b)
18
Q
Sin(x)
A
Cos(x)
19
Q
Sec(x)tan(x)
A
Sec^2(x)
20
Q
Sec^2(x)
A
2sec(x)sec(x)Tan(x)
21
Q
Sin(x)
A
Cos(x)
22
Q
Cos(x)
A
-Sin(x)
23
Q
-Csc(x)cot(x)
A
Csc(x)
24
Q
1/x
A
Ln(x)
25
A^xln(x)
A^x
26
Csc^2(x)
1+cot^2(x)
27
Sec^2(x)
Tan^2(x)+1
28
Csc(x)
1/sin(x)
29
d/dx tan^-1(u)
The derivative of the tangent inverse of x
(1/1+u^2)*(u’)
(1/1+u-squared)*(u’)
30
2(sinx)(cosx)
Sin(2x)
31
Sin(2x)
2(sinx)(cosx)
32
F(x)=sin^-1(3x)
This is a composite function where the inner function 3x is considered U
Find u’ first
33
When you see a function like this, what do you think?
Determine f’(x) of f(x)=sin^-1(3x)
-Composite function
-Which needs Chain Rule
- inner function =u - determine u’
-then inverse sin^-1(u) = 1/sqrt(1-u^2)
-multiple inverse sign by u’
=f’(x)=1/sqrt(1-3x^2)^2
=f’(x)=3/sqrt(1-9x^2)
34
What is the inner function of -3cos^4(x)?
U=cos(x)
35
What is the derivative of -3u^4?
-12u^3 multiplied by U’ (u prime)
36
What is u and u prime of f(x)=-3cos^4(x)?
U= cos(x)
U’ = -sin(x)
37
Derivative of sin (u) =
Cos(u) * u’
Cos(u) X u-prime
38
Derivative of cos(u)
d/dx[cos(u)] =
-Sin(u)*u’
Negative sin(u) multiplied by u-prime
39
d/dx tan(u)=
Sec^2(u)*u’
Sec^2(u) multiplied by u-prime
40
d/dx[csc(u)]=
Negative csc(u)cot(u)*u’
-csc(u)cot(u) multiplied by u-prime
41
d/dx[sec(u)]=
Sec(u)tan(u)*u’
42
d/dx [cot(u)]=
-csc^2(u)*u’
43
d/dx[e^u]=
E^u*u’
44
d/dx [u/v]=
vu’-uv’/v^2
45
d/dx[u*v]=
uv’+vu’
46
d/dx[u^n]=
n*u^n-1*u’
47
d/dx[sin^-1(u)]=
The derivative of sin inverse of u?
d/dx[sin^-1(u)]=1/sqrt(1-u^2)*u’
48
d/dx[cos^-1(u)]=
The derivative of cosine inverse of u?
d/dx[cos^-1(u)]= -1/sqrt(1-u^2) * (u’)
Negative 1 divided by the square root of 1 minus u squared * u prime
49
F(x)*g(x)=
F(x)g’(x)+g(x)f’(x)
50
F(x)/g(x)=
(g(x)f’(x)-f(x)g’(x))/g(x)^2
51
F(g(x)=
F’(g(x))g’(x)
Derivative of the outer function X Derivative of the Inner function
52
Power rule using the chain rule
d/dx[u^n]=nu^n-1•u’
53
d/dx[tan^-1(u)]=
d/dx[tan^-1(u)]= 1/(1+u^2)*(u’)
54
Quotient Rule
d/dx[f/g]=gf’-fg’/g^2
55
d/dx[cot^-1(u)]=
Derivative of Cotangent inverse u=
d/dx[cot^-1(u)]= -1/(1+u^2)*(u’)
Derivative of Cotangent inverse u= negative 1/(1+u-squared)*(u-prime)
56
Derivative of ln(x)
1/x
57
Derivative of secant inverse u=
d/dx[sec^-1(u)]=
d/dx[sec^-1(u)]= 1/ (/u/(sqrt(u^2-1)*(u’)
Derivative of secant inverse u = 1/(the absolute value of u)*sqrt(u-squared-1) *(u-prime)
58
Product Rule
h(x)=F(x)g(x)
2 functions being multiplied together
Copy 1st function (derivative of 2nd)
+ copy 2nd function (derivative of 1st)
h’(x)=f(x)g’(x)+g(x)f’(x)
59
Quotient Rule
Q(x)= B(x)T’(x)-T(x)B’(x)
————————
B(x)^2
60
Derivative of cosecant inverse u=
d/dx[csc^-1(u)]=
d/dx[csc^-1(u)]= -1/ (/u/(sqrt(u^2-1)*(u’)
= Derivative of cosecant inverse u = negative 1/(the absolute value of u)*sqrt(u-squared-1) *(u-prime)
61
62
63
64
Derivative of a function with a function in an exponent
Copy entire function * ln(base#)* derivative of whatever is in the exponent
65
What is the derivative of 4^7x-3
(4^7x-3)*ln(4)*7
Copy the function*ln(base#)*d’of expont
66
67
68
69
70
71
D/dx[y] =
The derivative of y with respect to x
Dy/dx
Derivative of y over the derivative of x
72
73
What is d/dx(log base 2(x))?
1/xln(2)
74
What is d/dx(a^x)?
A^x *(ln(a))
A to the x times the natural log of a
75
D/dx(a^x)?
A^x * (ln(a))
76
77
d/dx [ln(x^2+5)]=
1/( ) *(derivative of inner
1/(x^2+5)* (2x)
=2x/(x^2+5)
78
