unit 5 formula review sheet - Mrs. Price's Math Site - HOME€¦ · · 2016-01-04Log/Exponent...
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Log/Exponent Properties: ln(1) = 0 ln(e) = 1 ln(a n ) = n*ln(a) ln(ab) = ln(a) + ln(b) b a b a ln ln ln − = Exponent Properties: e a * e b = e a+b (e a ) b = e ab e 0 = 1 Log Differentiation steps: 1) Take ln of both sides 2) Expand right side. 3) Find derivative 4) Solve for dy/dx Evaluate derivative of inverse: (find !! ! () 1.Set f(x) = a and solve for x (guess and check) 2. Find f ‘(x) 3. Plug in x value from step #1 into f ‘(x). 4. Flip value. Log Derivatives: Exponential Derivatives d dx ln | u | = u ' u d dx e u = e u ∗ u ' u u a u dx d a ' * ln 1 log = d dx a u = ln a ∗ a u * u ' Trig Derivatives: d dx sin u = cos u * u ' d dx tan u = sec 2 u * u ' d dx sec u = sec u tan u * u ' d dx cos u = − sin u * u ' d dx cot u = − csc 2 u * u ' d dx csc u = − csc u cot u * Inverse Trig Derivatives: d dx arcsin u = u ' 1 − u 2 d dx arctan u = u ' 1 + u 2 d dx arc sec u = u ' u u 2 − 1 d dx arccos u = − u ' 1 − u 2 d dx arc cot u = − u ' 1 + u 2 d dx arc csc u = − u ' u u 2 − 1 Integral Formulas: Power Rule: u n du = u n+ 1 n + 1 + C ∫ Log Rule: 1 u du = ln | u | +C ∫ Exponential Rule: (Base e) ∫ du e u = e u +C Exponential Rule (base other than e) a u du = a u ln a + C ∫ *Note: lna is a constant* Trig Integrals: sin udu = − cos u + C ∫ cos udu = sin u + C ∫ sec 2 udu = tan u + C ∫ sec u tan udu = sec u + C ∫ csc 2 udu = − cot u + C ∫ csc u cot udu = − csc u + C ∫ C + | cosu | -ln tan = ∫ udu C u udu + = ∫ sin ln cot ∫ udu sec = ln|sec u + tan u| + C ∫ udu csc = ln|csc u + cot u| + C Inverse Trig Integrals: ∫ + = − C a u u a du arcsin 2 2 ∫ + = + C a u a u a du arctan 1 2 2 ∫ + = − C a u a a u u du | | sec arc 1 2 2 !"# ! ! = and log ! ! = log ! = ln ln Interest Formulas = !1 + ! (!") A = Pe rt
Transcript of unit 5 formula review sheet - Mrs. Price's Math Site - HOME€¦ · · 2016-01-04Log/Exponent...
Log/Exponent Properties: ln(1) = 0 ln(e) = 1 ln(an) = n*ln(a) ln(ab) = ln(a) + ln(b)
baba lnlnln −=⎟⎠
⎞⎜⎝
⎛
Exponent Properties: ea * eb = ea+b (ea)b = eab e0 = 1
Log Differentiation steps: 1) Take ln of both sides. 2) Expand right side. 3) Find derivative 4) Solve for dy/dx Evaluate derivative of inverse: (find 𝑓!! !(𝑎) 1.Set f(x) = a and solve for x (guess and check) 2. Find f ‘(x) 3. Plug in x value from step #1 into f ‘(x). 4. Flip value.
Log Derivatives: Exponential Derivatives ddxln | u |= u '
u
ddxeu = eu ∗u '
uu
au
dxd
a'*
ln1log =
ddxau = lna∗au *u '
Trig Derivatives: d
dxsinu = cosu*u '
ddxtanu = sec2 u*u '
ddxsecu = secu tanu*u '
ddxcosu = −sinu*u '
ddxcotu = −csc2 u*u '
ddxcscu = −cscucotu*u '
Inverse Trig Derivatives: ddxarcsinu = u '
1−u2 ddxarctanu = u '
1+u2 ddxarcsecu = u '
u u2 −1
ddxarccosu = − u '
1−u2 ddxarccotu = − u '
1+u2
ddxarccscu = − u '
u u2 −1
Integral Formulas: Power Rule:
un du = un+1
n+1+C∫
Log Rule: 1udu = ln | u |+C∫
Exponential Rule: (Base e) ∫ dueu = eu + C
Exponential Rule (base other than e)
au du = au
lna+C∫
*Note: lna is a constant*
Trig Integrals: sinudu = −cosu+C∫ cosudu = sinu+C∫ sec2 udu = tanu+C∫ secu tanudu = secu+C∫ csc2 udu = −cotu+C∫ cscucotudu = −cscu+C∫
C + |cosu|-lntan =∫ udu Cuudu +=∫ sinlncot
∫ udusec = ln|sec u + tan u| + C
∫ uducsc = -‐ln|csc u + cot u| + C
Inverse Trig Integrals:
∫ +=−
Cau
uadu arcsin
22 ∫ +=
+C
au
auadu arctan1
22
∫ +=−
Cau
aauudu ||secarc1
22
𝑎!"#! ! = 𝑥 and log! 𝑎! = 𝑥
log! 𝑥 = ln 𝑥ln𝑎
Interest Formulas
𝐴 = 𝑃 !1 +𝑟𝑛!(!")
A = Pert
