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#1 2006-03-29 10:06:10

MathsIsFun
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Registered: 2005-01-21
Posts: 7,535

Differential Calculus Formulas

Differential Calculus Formulas


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#2 2006-03-30 15:46:06

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Differential Calculus Formulas


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2006-03-31 15:41:58

George,Y
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Registered: 2006-03-12
Posts: 1,306

Re: Differential Calculus Formulas

--solving procedure

p(t) could any function without y, constant is ok.

Linear means no y, y[sup]2[/sup] or yy', etc.

So virtually there are only y and y' multiplied by function of t or constant,  function of t,  and constant are allowed in the 1st Order Linear DE. And the DEs satisfying this condition can be easily transformed into the standard form above through division.

Last edited by George,Y (2007-05-21 16:22:02)


X'(y-Xβ)=0

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#4 2006-04-08 03:24:54

ganesh
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Registered: 2005-06-28
Posts: 14,351

Re: Differential Calculus Formulas

Derivatives of some Elelmentary functions

Derivative of a constant is zero.

If u and v are functions of x, and c is a constant,

provided v≠0


Character is who you are when no one is looking.

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#5 2006-04-08 03:38:37

ganesh
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Registered: 2005-06-28
Posts: 14,351

Re: Differential Calculus Formulas

Chain rule

Inverse Function


Character is who you are when no one is looking.

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#6 2006-04-08 03:48:07

ganesh
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Registered: 2005-06-28
Posts: 14,351

Re: Differential Calculus Formulas

Inverse Trignometric Functions


Character is who you are when no one is looking.

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#7 2006-04-21 17:34:59

ganesh
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Registered: 2005-06-28
Posts: 14,351

Re: Differential Calculus Formulas

nth derivative



Leibnitz theorem
If u and v are functions of x and n is a positive integer, then



Character is who you are when no one is looking.

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#8 2006-05-05 21:49:38

ganesh
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Registered: 2005-06-28
Posts: 14,351

Re: Differential Calculus Formulas

Hyperbolic functions

Inverse Hyperbolic functions


Character is who you are when no one is looking.

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#9 2006-05-05 23:50:10

ganesh
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Registered: 2005-06-28
Posts: 14,351

Re: Differential Calculus Formulas

Derivatives of functions of the form (x² ± a²)


Character is who you are when no one is looking.

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#10 2006-05-16 02:02:47

ganesh
Moderator
Registered: 2005-06-28
Posts: 14,351

Re: Differential Calculus Formulas

Tangents and Normals

If y=f(x) is a curve,

If dy/dx=0 at a point P, then the slope of the tangent at P is zero, therefore, the tangent is parallel to the x-axis.

If dy/dx = infinity at a point P, then the slope of the tangent at P is infinity, therefore, the tangent is parallel to the y-axis.

Equations of tangent and normal at a point:-

Slope of the tangent at point P (x1, y1) on the curve y=f(x) is the value of dy/dx at (x1,y1). Let dy/dx=m at (x1,y1).
Then the equation of the tangent at (x1,y1) is

Normal at point P is perpendicular to the tangent at P and passes through P(x1,y1).
Therefore,

Equation of the normal at P(x1,y1) is

Angle between two curves

The angle between two curves is the angle between the tangents to the curves at the point of intersection.

Let

be the equations of two curves C1 and C2 intersecting at P.

Then, the slope of tangent at P to the first curve is given by

at P.

Similarly, the slope of the tangent to the second curve is given by

at P.

Then, the angle between the curves is given by

or

If m1=m2, then the curves touch each other at P.
If m1m2=-1, then the curves cut each other orthogonally at P.


Character is who you are when no one is looking.

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#11 2006-08-05 18:32:28

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: Differential Calculus Formulas

Derivative of a Parametric Function

The derivative of a parametric function defined by x = f(t) and y = g(t) is given by

L'Hôpital's Rule

For differentiable functions f(x) and g(x),

This is useful for determining limits which give indeterminate forms such as 0/0, ∞/∞, 0[sup]0[/sup], 1[sup]∞[/sup], 0 × ∞, ∞ - ∞, and ∞[sup]0[/sup].

Local Extrema (Maxima and Minima)

The point a on a curve f(x) is a local maximum if f'(a) = 0 and f''(a) < 0.

The point a on a curve f(x) is a local minimum if f'(a) = 0 and f''(a) > 0.

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#12 2006-08-06 22:00:43

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: Differential Calculus Formulas

The Laplace Transform

The Laplace transform is a useful tool for solving differential equations. The Laplace transform F(s) of f(x) is defined by

The Laplace transform is a linear operator, and thus

The inverse Laplace transform

is also a linear operator:

Solution of a Linear Differential Equation of Constant Coefficients using a Laplace Transform

Consider the linear differential equation

Taking the Laplace transform of both sides, and noting that

we may write the differential equation as



or, collecting coefficients,




Now, given that we have the initial conditions at x = 0, we can proceed to solve for L[y], and once we have an equation for L[y], we may simplify it (often by means of partial fractions) and then find the inverse Laplace transform, and thus find y. If initial conditions are given at some x[sub]0[/sub] ≠ 0, let  u = x - x[sub]0[/sub] and solve the differential equation in terms of u, then substitute the value of x back in when the solution is finished.

List of Selected Laplace Transforms

Note: Taking the inverse Laplace transform of these equations will give an expression for the inverse Laplace transform of a function F(s).











Feel free to request any other Laplace transforms/inverse Laplace transforms. I have about 150 other transforms for a wide variety of cases, but they seem too obscure for posting.

Example problems may be posted on request to aid in clarity.

Last edited by Zhylliolom (2006-08-06 22:03:52)

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#13 2008-12-30 07:10:19

Manta
Member
Registered: 2008-12-30
Posts: 1

Re: Differential Calculus Formulas

lol:lol::lol: i am very obseded by those functions[integral,derivative,trigonometry,etc... even if i have some difficulties in some of them] and calculus even if i have difficulties understanding some of them(i am only 12 years old).I also loved quadratic equations when i was younger. roflol

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#14 2010-05-20 04:27:26

wailinaung
Member
Registered: 2010-05-20
Posts: 1

Re: Differential Calculus Formulas

differentional equation problems  y'=x^x   how to solve? and y'=(arctgx)^x  how to solve?

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#15 2010-05-20 09:22:16

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Differential Calculus Formulas

Hi wailinaung;

y'=x^x  Since the variables are already separated for you you just need to integrate both sides. Offhand I do not know how to integrate x^x. Same goes for integrating arctan(x)^x.

Checking pierce's table of integrals neither of them is there. Googling says they cannot be integrated in terms of elementary functions. So solving those DE's is out. You can get an approximation to them by series methods and also if you are given a numerical initial condition, by numerical methods.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#16 2010-09-21 03:19:32

hariprasath2
Member
Registered: 2010-09-21
Posts: 1

Re: Differential Calculus Formulas

ganesh wrote:

Derivatives of some Elelmentary functions

Derivative of a constant is zero.

If u and v are functions of x, and c is a constant,

provided v≠0

:(thank u ganesh
up

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