Formula Sheet for Final Exam
Differentiation Rules
′
⎛ f ⎞ g⋅ f′− f ⋅g′
Product: (f ⋅g)′= f ⋅g′+g⋅ f′ Quotient: ⎜⎜ ⎟⎟ = 2
⎝ g ⎠ g
Exp, Log Derivatives
d x x d 1
a = a lna (loga x) =
dx dx xlna
Trig Derivatives
d d d 2
sinx =cosx cosx −sinx tanx =sec x
dx dx dx
d 2 d d
cotx = −csc x secx =secxtanx cscx =−cscxcotx
dx dx dx
d −1 1 d −1 1 −1 1
(sin x)= (cos x)= − (csc )=−
dx 2 2 2
1−x dx 1−x x −1
d −1 1 d −1 1 d −1 1
(sec x)= (tan x)= (cot x)=−
dx 2 2 2
x x −1 dx 1+ x dx 1+ x
Hyperbolic Functions
x −x x −x
e −e 1 e +e
sinhx= cschx= coshx=
2 sinhx 2
1 sinhx coshx
sechx= tanhx= cothx=
coshx coshx sinhx
Derivatives of Hyperbolic Functions
d d
(sinhx)=coshx (cschx)=−cschx⋅cothx
dx dx
d d
(coshx)=sinhx (sechx)=−sechx⋅tanhx
dx dx
d 2 d 2
(tanhx)=sech x (cothx)=−csch x
dx dx
Derivatives of Inverse Hyperbolic Functions
d −1 1 d −1 1 d −1 1
(sinh x)= (cosh x)= (tanh x)=
dx 2 2 2
1+x dx x −1 dx 1−x
Table of Integrals
n+1
n x x x
x dx= +C , (n ≠ −1) e dx = e +C
∫ n+1 ∫
x
x a 1
a dx = +C, a ≠1 dx = ln| x |+C
∫ lna ∫ x
sin(x)dx=−cosx+C cos(x)dx =sin x+C
∫ ∫
2 2
sec (x)dx=tanx+C csc (x)dx=−cotx+C
∫ ∫
sec(x)tan(x)dx=secx+C csc(x)cot(x)dx=−cscx+C
∫ ∫
Summation Formulas
n n n(n +1)
c = nc
∑ i
∑ =
i=1 i=1 2
n n 2
2 n(n +1)(2n +1) 3 ⎡n(n +1)⎤
i
∑ = i =
∑ ⎢ ⎥
i=1 6 i=1 ⎣ 2 ⎦
Miscellaneous Formulas
st f (x+h)− f (x)
1 Principles Derivatives: f ′(x) = lim
h→0 h
Linear Approximation: f (x) ≈ f (a) + f ′(a)(x − a)
f (b)− f (a)
Mean Value Theorem: f ′(c) =
b − a
b
1
Average Value: fave = f (x)dx
b − a ∫
a
b
Mean Value Theorem for Integrals: f (x)dx = f (c)(b−a)
∫
a
• NOTE: If needed, the formulas for the inverse hyperbolic functions (e.g.
−1 2
sinh x=ln(x+ x +1)) will be provided within the question. In addition, any
formulas for surface area and volume of geometric shapes (which sometimes
come up in related rates/optimization questions) will be provided.