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Calculus Basic Differentiation Rules
Basic Differentiation/Integration Formal Rules with follow up example
41
Mathematics
Undergraduate 2
07/06/2019
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Term
d/dx [|u|] = ?
 Definition
d/dx [|u|] = (u/|u|)(u'), u ≠ 0
Term
d/dx [ln u] = ?
 Definition
d/dx [ln u] = u'/u
Term
d/dx [ln(5x3)] = ?
 Definition
d/dx [ln(5x3)] = 15x2/5x3 = 3/x
Term
d/dx [eu] = ?
 Definition
d/dx [eu] = (eu)u'
Term
d/dx[e(5x^3)] = ?
 Definition
d/dx [e5x^3] = (e5x^3)(15x2)
Term
d/dx[logau] = ?
 Definition
d/dx[logau] = u'/(ln a)u
Term
d/dx[log520x3] =
 Definition
d/dx[log520x3] = 60x2/(ln 5)20x3 = 3/(x(ln 5))
Term
d/dx [sec u] = ?
 Definition
d/dx [sec u] = (sec u tan u)u'
Term
d/dx [sec 2x8] = ?
 Definition
d/dx [sec 2x8] = (sec 2x8 tan 2x8)16x7
Term
d/dx[csc u] = ?
 Definition
d/dx[csc u] = -(csc u cot u)u'
Term
d/dx[csc 3x2] = ?
 Definition
d/dx[csc 3x2] = -(csc 3x2 cot 3x2)6x
Term
d/dx[au] = ?
 Definition
d/dx[au] = (ln a)au u'
Term
d/dx[53x^3] = ?
 Definition
d/dx[53x^3] = (ln 5)(53x^3)9x2
Term
∫eudu = ?
 Definition
∫eudu = eu+C
Term
∫2e2xdx = ?
 Definition
∫2e2xdx = e2x + C
Term
∫audu = ?
 Definition
∫audu = au(1/ln(a)) + C
Term
∫55xdx = ?
 Definition
∫55xdx = (1/5) (55x) (1/ln(5)) + C
Term
∫tan(u)du = ?
 Definition
∫tan(u)du = -ln|cos(u)|+C
Term
∫tan(5x)dx = ?
 Definition
∫tan(5x)dx = -(1/5)ln|cos(5x)| + C
Term
∫cot(u)du = ?
 Definition
∫cot(u)du = ln|sin(u)|+C
Term
∫cot(8x)dx = ?
 Definition
∫cot(8x)dx = (1/8)ln|sin(8x)| + C
Term
∫sec(u)du = ?
 Definition
∫sec(u)du = ln|sec(u) + tan(u)| + C
Term
∫sec(25x)dx = ?
 Definition
∫sec(25x)dx = (1/25) ln|sec(25x) + tan(25x)| + C
Term
∫csc(u)du = ?
 Definition
∫csc(u)du = -ln|csc(u) + cot(u)| + C
Term
∫csc(2x)dx = ?
 Definition
∫csc(2x)dx = (1/2) -ln|csc(2x) + cot(2x)| + C
Term
∫sec(u) tan(u) du = ?
 Definition
∫sec(u) tan(u) du = sec(u) + C
Term
∫sec(51x) tan(51x) dx = ?
 Definition
∫sec(51x) tan(51x) = (1/51) sec(51x) + C
Term
∫csc(u) cot(u) du = ?
 Definition
∫csc(u) cot(u) du = -csc(u) + C
Term
∫csc(16x) cot(16x) dx = ?
 Definition
∫csc(16x) cot(16x) dx = (1/16) - csc(16x) + C
Term
logaxy = ?
 Definition
logaxy = logax + logay
Term
logax/y = ?
 Definition
logax/y = logax - logay
Term
loga1/x = ?
 Definition
loga1/x = -logax
Term
logaxy = ?
 Definition
logaxy = y logax
Term
What is the disk method for volumes of revolution around the horizontal axis?
 Definition
V=π∫x2dy
Term
Think about how to come up the area of a circle by cutting the circle into a pie shape and turning it into a rectangle
 Definition
https://www.google.com/search?q=understanding+area+of+a+circle&rlz=1C1CHBF_enUS849US849&oq=understanding+area+of+a+&aqs=chrome.0.0j69i57j0l4.3823j1j7&sourceid=chrome&ie=UTF-8#kpvalbx=_iIpaXaWyHo-zggfvkbHQAQ21
Term
What is the shell method to finding the volume through integration around the vertical axis?
 Definition
V=2π∫xy dy.  The equation must be substituted into the x value so that everything can be integrated with respect to y.
Term
∫du/(a^2+u^2) = ?
 Definition
(1/a)arctan(u/a) + C
Term
cos2A = ?
 Definition
cos2A = 1/2 + (cos(2A))/2
Term
sin2A = ?
 Definition
sin2A = 1/2 - (cos(2A))/2
Term
1+tan2x = ?
 Definition
1+tan2x = sec2x
Term
1+cot2x = ?
 Definition
1+cot2x = csc2x
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