- In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for re-election. (Hugo Rossi)
- Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be. (Bertrand Russell)
- Who has not been amazed to learn that the function , like a phoenix rising from its own ashes, is its own derivative? (Francois le Lionnais)
- I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives.(Charles Hermite)
- Senate Panel Approves Tougher Rules on Derivatives.
Derivative with respect to x
We now extend to a function u(x) depending on position instead of time t.
Definition: A function where , said to be differentiable in with derivative or gradient
if for some positive constant and all ,
In particular, choosing , we have
- for small,
which means that is the derivative of f(x) with respect to , with and kept constant, referred to as the partial derivative with respect to .
The definition directly generalises to real-valued function u(x) of d-vector variable , where the variable components can have have some other meaning than position. In the case , that is with u(x) a function of one variable , we sometimes use to denote the derivative (following Newton), thus with the defining relation
- for small.
Seeking the derivative as the slope of the tangent for a function f(x) of one real variable x.
Vector-valued function of vector variable
The definition of derivative directly generalize to an m-vector-valued function of an n-vector variable :
Definition: A function is differentiable with derivative (or ) if for som positive constant
Here the derivative is an matrix.
We shall use this derivative below when solving an equation where is a differentiable function with non-singular derivative using Newton’s method.