Basic Calculus Flashcards
d/dx (sin 5x)
5 cos 5x
d/dx (tan 9x)
9 sec^2 (9x)
d/dx (cot 7x)
-7 csc^2 (7x)
d/dx (csc x^2)
-2x csc(x^2) cot(x^2)
Quotient Rule
lo d’hi - hi d’lo/ (lo)^2
Product Rule
1st d’2nd + 2nd d’1st
d/dx (cos u)
-sin u du/dx
d/dx (sin u)
cos u du/dx
d/dx (tan u)
sec^2 u du/dx
d/dx (cot u)
-csc^2 u du/dx
d/dx (sec u)
sec u tan u du/dx
If f(x) = c where c is a constant
then f’(x) = 0
Constant Rule
If f(x) = x, then f’(x) = 1
Identity Rule
If f(x) = x^n, then f’(x) = nx^n-1
Power Rule
If f(x) = Cx^n, then f’(x) = Cnx^n-1
Constant Multiple Rule
If f(x) = u(x) +/- v(x)
then f’(x) = u’(x) +/- v’(x)
Sum and Difference Rule
If f(x) = u(x) • v(x)
then f’(x) = u(x) - v’(x) + v(x) - u’(x)
Product Rule
If f(x) = u(x)/ v(x)
then f’(x) = v(x) - u’(x) - u(v) • v’(x)
/ [u(x)]^2
Quotient Rule
If f(x) = [u(x)]^n
then f’(x) = n [u(x)] ^ n-1 [u’(x)]
Chain Rule
e^5x
5e^5x
d/dx (e^2x)
2e^2x
d/dx (e^cos 3x)
u = cos 3x
cos (5x^2)
u = 5x^2
du = 10x
sin (3x^2)
u = 3x^2
du = 6x
tan (5x)
u = 5x
du = 5
tan ( x+3 )
u = x+3
du = 1
tan ( 2x^3 )
u = 2x^3
du = 6x^2
u = cos 3x
du = ?
du = -3 sin 3x
d/dx ( xe^2x )
x • 2e^2x + e^2x • 1
ANS: ?
ANS: 2xe^2x + e^2x
u = 2x^2 + x
du = ?
du = 4x + 1
u = x^2
du = ?
du = 2x
u(x) = sin e^3x
u’(x) = ?
u’(x) = (cos e^3x) (3e^3x)
u(x) = sin (3x^2)
u’(x) = ?
u’(x) = 6x cos (3x^2)
u(x) = tan (5x)
u’(x) = ?
u’(x) = 5 sec^2 (5x)
u(x) = tan (x+3)
u’(x) = ?
u’(x) = 1 sec^2 (x+3) or sec^2(x+3)
u(x) = sin (2x)
u’(x) = ?
u’(x) = 2 cos (2x)
u(x) = sin ( 4x )
u’(x) = ?
u’(x) = 4 cos (4x)
u = sin 3x
u’ = ?
u’ = 3 cos 3x
Product Rule: 1st • d’2nd + 2nd • d’1st
u(x) = 4x^2
u’(x) = 8x
v(x) = tan (2x^3)
v’(x) = 6x^2 sec^2 (2x^3)
ANS: ?
ANS: 4x^2
( 6x^2 sec^2 (2x^3) )
+ tan (2x^3) • 8x
Product Rule: 1st • d’2nd + 2nd • d’1st
u(x) = sin (2x)
u’(x) = 2 cos (2x)
v(x) = cos (3x)
v’(x) = -3 sin (3x)
ANS: ?
ANS: sin (2x) [ -3 sin (3x) ]
+ cos (3x) [ 2 cos (2x) ]
e^u • du
e^cos3x • -3 sin 3x
ans: ?
ans: -3e^cos3x sin 3x
or -3 sin 3x (e^cos3x)
a^u ln a • du
Dx (10^x) = ?
10^x ln 10 • 1
a = 10
u = x
du = 1
Quotient Rule: lo d’hi - hi d’lo / (lo)^2
d/dx ( x/ 10^x + 8 )
ANS: ?
Dx(10^x) = 10^x ln 10
ANS: ( 10^x + 8 ) (1) - x ( 10^x ln 10 )
/ ( 10^x + 8 ) ^2
Dx ( ln u ) = ?
u’ / u
ln u^2
2 (ln u)
log 10 x^2
2 ( log 10 x )
x^2 ( ln x^5 )
x^2 ( 5 ln x )
ln ( x - 5 ) ^1/2 - ln ( x^2-7 )
1/2 ln ( x-5 ) - ln ( x^2 - 7 )
d/dx (a^u) = ?
a^u ln a du/dx
sin x
1 / csc x
csc x
1 / sin x
cos x
1 / sec x
sec x
1 / cos x
tan x
1 / cot x
cot x
1 / tan x
tan x
sin x / cos x
cot x
cos x / sin x
? = 1
sin^2 x + cos^2 x
? = sec^2 x
tan^2 x + 1
? = csc^2 x
1 + cot^2 x
