Definition:Derivative/Real Function/Derivative at Point/Definition 2

Definition

Let $I$ be an open real interval.

Let $f: I \to \R$ be a real function defined on $I$.

Let $\xi \in I$ be a point in $I$.

Let $f$ be differentiable at the point $\xi$.

That is, suppose the limit $\ds \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$ exists.

Then this limit is called the derivative of $f$ at the point $\xi$.

Also see

• Equivalence of Definitions of Derivative

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.2$ Derivatives: Definition $2$
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 10.1$
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $8$: The System of the World: Calculus
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): derivative