# Derivative with respect to Space

*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 depending on position instead of time .

**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

- .

which means that is the derivative of with respect to , with and kept constant, referred to as the *partial derivative* with respect to .

The definition directly generalizes to real-valued function of d-vector variable , where the variable components can have have some other meaning than position. In the case , that is with latex $u(x)$ a function of one variable ,we often use to denote the derivative, thus with the defining relation

- .

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 -vector-valued function of an -vector variable :

**Definition**: A function is differentiable with derivative if for som positive constant

- .

Here the derivative is an *matrix.*

We shall use this derivative below when solving an equation where $u:R^n\rightarrow R^n$ is a differentiable function with non-singular derivative using *Newton’s method.*

# Read More

- \hyperref[sevvar]{Calculus of Several Variables.}

Charles Babbage’s Analytical Engine 1871

Next: Rules of Differentiation Previous: Derivate with respect to Time.

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