Calc 2 Notes
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Product Rule:​ (f g)′ = f ′g + f g ′
( )′ =
f
g
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f ′g−f g ′
g2
sin(cos−1 x) = √1 − x2
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Quotient Rule:
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Dx ln u = u1 Dx u
sec(tan−1 x) = √1 + x2
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∫ u1 du = ln |u| + C
tan(sec−1 x) = √x2 − 1, if x ≥ 1
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ln1=0
cos(sin−1 x) = √1 − x2
− √x2 − 1, if x ≤− 1
Dx sinu = cosu · u′ Dx cosu =− sinu · u′
ln ab =ln a +ln b
ln ba =ln a -ln b
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Dx tanu = sec2 u · u′
ln ar = r ln a
Logarithmic differentiation helps with a/b or a · b.
​exists and has an inverse.
1
(f )′(y) = f (x)
′
If​ f ′(x) =/ 0 , f
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●
−1
elnx = x
u
ln(ex ) = x
u
∫
eu du
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Dx e = e Dx u
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ln (ax ) = ln (exlna ) = x ln a
y
ax ay = ax+y (ax ) = axy
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x
( ab ) =
x
(ab) = ax bx
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Dx secu = secutanu · u′ Dx cscu =− cscucotu · u′
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−1
=
eu
1
· u′, − 1 ≺ u ≺ 1
√1−u2
Dx cos−1 u =− 1 2 · u′, − 1 ≺ u ≺ 1
√1−u
1
−1
Dx tan u = 1+u2 · u′
Dx sin−1 u =
Dx sec−1 u =
+C
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2
2
Dx au = au ln a · Dx u
2
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xx = exlnx Dx = exlnx DX (xlnx)
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=
x · 1x +
lnx
log a x = lna
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(
∫ xa dx =
xx (1
+ lnx)
lnx) =
1
Dx log a x = xlna
x
+C
a+1
●
dy
Exponential Growth/Decay: dt =
dT
Newton Cooling:​ dt = k (T − T 1 )
Continuous:​
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a+1
Compound Interest:​
ert
x = sin−1 y ⇔ y = sinx, ​
y = y 0 ekt
ky
A(t) = A0 (1 +
A(t) = A0
−π
2
≤x≤
−π
2
≤x≤
π
2
x = sec−1 y ⇔ y = secx, 0 ≤ x ≤ π, x =/
2
1
−1
a sec
ex −e−x
2
sinhx
tanhx = coshx
1
sechx = coshx
|x|
a
coshx =
sinhx =
2
( )+C
​
cothx =
cschx =
2
ex +e−x
2
coshx
sinhx
1
sinhx
cosh x − sinh x = 1
Dx sinhu = coshu · u′
Dx coshu = sinhu · u′
Dx tanhu = sech2 u · u′ DX cothu =− csch2 u · u′
Dx sechu =− sechutanhu · u′
r nt
)
n
Dx cschu =− cschucothu · u′
π
2
x = cos−1 y ⇔ y = cosx, 0 ≤ x ≤ π
x = tan−1 y ⇔ y = tanx, ​
dx = a1 tan−1 ( ax) + C
∫ x√x1−a dx =
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· u′, |u| ≻ 1
2
2
xx
1
|u|√u2 −1
∫ √a1−x dx = sin−1 ( ax) + C
1
∫ a +x
ax
bx
1
∫ ax dx = ( lna
) ax + C
Dx cotu =− csc2 u · u′
π
2
Absolute Value Derivative: |u(x)|’ = [ u(x) / |u(x)| ]*u’(x)
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Dx sinh−1 x =
1
√x2 +1
Dx cosh−1 x =
1
, x≻1
√x2 −1
1
Dx tanh−1 x = 1−x
2, − 1 ≺ x ≺ 1
Dx sech−1 x =
−1
,
x√1−x2
0≺x≺1
Trig Identities
First-Order Linear EQ:
dy
dx
+ P (x)y = Q(x)
Reciprocals:
∫ P (x)dx
1)Find Integrating Factor
e
2) Multiply both sides by
e
1
sinu = cscu
1
tanu = cotu
∫ P (x)dx
Pythagorean Identities:
sin2 u + cos2 u = 1
1 + tan2 u = sec2 u
3) The equation becomes:
∫ P (x)dx
dy
dx
e
∫ P (x)dx
+e
∫ P (x)dx
P (x)y = e
4)Left side is the derivative of the product
So the equation becomes:
∫ P (x)dx
d
dx
y·e
∫ P (x)dx
=e
1
cosu = secu
​
Q(x)
∫ P (x)dx
y·e
1 + cot2 u = csc2 u
Quotients:
sinu
tanu = cosu
​
cotu = cosu
sinu
Sum/Difference Formulas:
sin(u + v ) = sinucosv + cosusinv
sin(u − v ) = sinucosv − cosusinv
Q(x)
cos(u + v ) = cosucosv − sinusinv
cos(u − v ) = cosucosv + sinusinv
tanu+tanv
tan(u + v ) = 1−tanutanv
5) Integrate both sides:
∫ P (x)dx
ye
∫ P (x)dx
= ∫ Q(x)e
dx
−∫ P (x)dx
∫
Double Angle Formulas:
sin(2u) = 2sinucosu
6) Solution is:
y=e
tanu−tanv
tan(u + v ) = 1+tanutanv
cos(2u) = cos2 u − sin2 u
∫ P (x)dx
Q(x)e
dx
= 2cos2 u − 1
= 1 − 2sin2 u
2tanu
tan(2u) = 1−tan
2u