Formula Sheet for Final Exam
Inverse Trig Derivatives
d
1
(sin 1 x ) 
dx
1 x2
d
1
(sec 1 x ) 
dx
x x2 1
d
1
(cos 1 x )  
dx
1 x2
d
1
(tan 1 x ) 
dx
1 x2
d
1
(csc 1 x )  
dx
x x2 1
d
1
(cot 1 x )  
dx
1 x2
Derivatives of Hyperbolic Functions
d
d
(sinh x)  cosh x
(csc hx)   csc hx  coth x
dx
dx
d
d
(sec hx)   sec hx  tanh x
(tanh x)  sec h 2 x
dx
dx
Derivatives of Inverse Hyperbolic Functions
d
1
d
1
(sinh 1 x) 
(cosh 1 x) 
dx
dx
1 x2
x 2 1
d
(cosh x)  sinh x
dx
d
(coth x)   csc h 2 x
dx
d
1
(tanh 1 x) 
dx
1 x2
Table of Integrals
ax
 a dx  ln a  C , a  1
 x dx  ln | x | C
 cos( x)dx  sin( x)  C
 sec ( x)dx  tan( x)  C  csc
 csc( x) cot( x)dx   csc( x)  C
1
x
 sin( x)dx   cos( x)  C
2
 sec( x) tan( x)dx  sec( x)  C
2
( x)dx   cot( x)  C
Miscellaneous Formulas
A Special Trig Limit: lim
0
Linear Approximation:
sin 
1

MVT: f (c) 
f (b)  f (a)
ba
L( x)  f (a)  f (a)( x  a)
n ( n  1)
Summation:  i 
2
i 1
n
n(n  1)( 2n  1)
i 

6
i 1
n
2
 n(n  1) 
i 

 2 
i 1
n
2
3
n
Riemann Sum (definition of definite integral):
lim
n 
 f ( x *)x
i 1
b
b
Average Value: f ave 

1
f ( x)dx
b  a a
i
MVT for Integrals:
 f ( x)dx  f (c)(b  a)
a
NOTES TO STUDENTS (this 2nd page will not be
included at the exam…it’s just for your info now):

If needed, the formulas for the hyperbolic and inverse hyperbolic functions (e.g.
e x  ex
or sinh 1 x  ln( x  x 2  1) ) will be provided within the
sinh x 
2
question. We’d also provide lim cos   1  0 if needed. In addition, any formulas
 0

for surface area and volume of particularly complicated geometric shapes (which
sometimes come up in related rates/optimization questions) will be provided.


You’ll note that we haven’t given you formulas for the derivatives of a x , sin( x ) ,
etc. Of course, you can use the “Table of Integrals” above to help you figure
these out. For example, since
d
sin( x)  cos( x) 
dx
 cos( x)dx  sin( x)  C , this means