Differentiating once gives you the gradient function. What do you get when you differentiate twice?
No, differentiating once gives you the differential, a row vector $Df(x)$.
The transpose of that vector is the gradient, $\nabla f(x)=Df(x)^\top$ (except in physics, numerics, optimization, where the derivative is sometimes denoted with the inverted triangle). Note that in generalized situations, transposition involves the coefficients of the scalar product.
Differentiating twice gives the second derivative, which is a tensor of order 2. Its coefficents can be arranged in matrix form, this is then called the Hesse or Hessian matrix.
The higher derivatives have no extra name.
First and second derivative are frequently used to form a quadratic approximation of the function. This is useful to compute directions of descend and testing for local minimality.
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TheEconomist
Updated on January 04, 2020Comments

TheEconomist almost 4 years
I understand why you would differentiate once  to get the gradient at a certain point.
But, what do you get when you differentiate once more like f''(x) or f'''(x).
I don't know how to visualize f''(x) or f'''(x).