Derived second

The derived second is the Dérivée derivative from a function, when it is defined.

Function of only one real variable

If the function admits a derivative second and that this derivative second is continuous, the function is known as of class C2.

Notation

If one notes ƒ ( X ) the function, then

  • the derivative is noted ƒ '( X ) or \ frac {df} {dx} , and
  • the derivative second is noted ƒ ( X ) (“F second of X”) or \ frac {d^2f} {dx^2}

Chart

The derivative second indicates the variation of the slope:

  • if it is positive on an interval, the slope increases, the curve is to the top, the function is known as “convex” on this interval;
  • if it is negative on an interval, the slope decreases, the curve is to the bottom, the function is known as “concave” on this interval;
  • if it is null, the curve is locally rectilinear;
  • if the derivative second is cancelled and changed sign, one has a Point inflection, the curve of the curve is reversed.

Function not admitting a derivative second

  • the nonderivable functions in a point do not admit there a derivative second; a fortiori noncontinuous functions in a point;

  • a Primitive of a function continues nonderivable is a continuous and derivable function, but it does not have a derivative second at the points where the initial function is not derivable; it is in particular the case of the primitive of primitive of a noncontinuous but limited function.
    • a double primitive of the function signs, ∫∫sgn
      \ sgn (X) = \ left \ {\ begin {matrix} -1 &: X < 0 \ \ \; 0 &: X = 0 \ \ \; 1 &: X > 0 \ end {matrix} \ right.
    • a primitive doubles of a square function, the primitive of a triangular function (in teeth of saw);
    • a primitive doubles whole function left ∫∫ E ( T )· dt ;
    • a primitive doubles decimal function left ∫∫ ( X - E ( T ))· dt ;

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