# Homotopy Type Theory partial derivative > history (Rev #2, changes)

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# Contents

## Definition

Given a calculus field?sequentially Cauchy complete Archimedean ordered field  F \mathbb{R} of scalars and a type of indices $I$, one could define a calculus vector space?real vector space  V \coloneqq F^I \mathbb{R}^I with a basis vector function $e:I \hookrightarrow V$. Let  f:V \to F \mathbb{R} be a differentiable scalar function, and given an index $i:I$, the partial derivative $\partial_{i}$ is pointwise defined as

$\partial_{i}(f)(v) \coloneqq \lim_{(x, y) \to (x, x)} \frac{f(v + x e_i) - f(v + y e_i)}{x - y}$