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McGill University

Mathematics & Statistics (Sci)

MATH 264

Iforgot

Fall

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Table of Common Derivatives
By David Abraham
Product and Quotient Rules:
d [ f x g x = d [f x g x + f x( ) d [g(x)]
dx dx dx
d d
d f x [ ( )] ( ) f ( ) [g(x)]
= dx 2 dx
dx x( ) g(x)
Trigonometric Functions:
d d d
sin(x)= cos(x) cos(x)= sin(x) tan(x)= sec (x)
dx dx dx
d d d 2
dxsec(x) =sec(x)tan(x) dx csc(x) = csc(x)cot(x) dx cot(x)= csc (x)
Inverse Trigonometric Functions:
d sin (x) = 1 d cos (x) = 1 d tan ( ) = 1
dx 2 dx 2 dx 1+ x 2
1 x 1 x
d sec ( ) = 1 d csc ( ) = 1 d cot ( ) = 1
dx x x 1 dx x x 1 dx 1+ x 2
Exponential and Logarithmic Functions:
d x x d x x d 1 d 1
a = a ln(a) e = e ln( ) = loga( x) =
dx dx dx x dx x ln( ) Table of Common Integrals
Standard Integration Techniques:
Integration by parts: f (x)g(x)dx = f (x) g(x)dx d [f x)] g x dx x
dx
2 2 2 a 2 2 2 a
Trig Substitution: a b x x = b sin() b x a x = b sec()
2 2 2 a
a + b x x = tan()
b
Trigonometric Identities:
Pythagorean Identities:
sin ( + cos ( = 1 1+ tan () = sec () 1+ cot () = csc ()
Sum-Difference Formulae:
sin(u v) = sin(u)cos(v) cos(u)sin(v) cos(u v) = cos(u)cos(v) sin(u)sin(v)
Half Angle and Power Reducing Formulae:
2 1 cos(2) 2 1+ cos(2)
sin ( = 2 cos ( = 2
2
sin(2) = 2sin()cos() cos(2)=12sin ()
Trigonometric Functions:
sin(u)du = cos(u) + c cos(u)du = sin(u) + c tan(u)du = lnsec(u) + c
2 2
sec (u)du = tan(u) + c csc (u)du = cot(u) + c sec(u)tan(u)du = sec(u) + c
csc(u)cot(u)du = csc(u) + c sec(u)du = lnsec(u) + tan(u) + c
csc(u)du = lncsc(u) cot(u) + c
Exponential and Logarithmic Functions:
u
e du = e + c a du = a +c ln(u)du = uln(u) u + c
ln( ) Inverse Trigonometric Functions:
1 1 1 1 1
2 2 du =sin a + c 2 2du = a tan a + c
a u a +u
1 1 1
du = sec + c
u u 2 a2 a
Advanced Calculus
Double Integrals:
The volume of a region D, bounded above by the function f(x,y) can be calculated via the
following double integral in Cartesian coordinates:
= , = ,
The equivalent in polar coordinates is then:
= , = ,
Where:
= +
=
=
If the area of the region D is desired instead of the volume of the region, the integrand f(x,y) or
f(r,) = 1.
Triple Integrals:
Triple integrals over a three dimensional region can be computed in the following way:
,, = ,,
The volume of the region can be computed if the integrand is made to be 1. The equivalent in Cylindrical Coordinates is then:
,, = ,,
Where:
= +
=
=
=
The equivalent in Spherical Coordinates is then:
,, = ,, sin
Where:
= sin cos
= sin sin
= cos
= + +
Change of Variable: The Jacobian
When performing a double or triple integral, sometimes the region D or V can be simplified by
use of a change of variable. In such a case, a transformation factor must be added to the
integration. This factor is equal to the determinant of the Jacobian matrix:
,
= =
,
Therefore:
=
Vector Fields:
A vector field is a function that assigns a vector, having both magnitude and direction to any
point in 2 or 3 dimensional space. It has the general form in three dimensions:
= ,, + ,, + ,,

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