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

Inverse Trignometric Functions

Hyperbolic functions

Inverse Hyperbolic functions

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.

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 x₀ ≠ 0, let  u = x - x₀ 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 (2006-08-07 20:03:52)

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

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