Derivative

Definition at a point

Algebraic definition

Suppose $f$ is a function defined on a subset of the reals and $x_0$ is a point in the interior of the domain of $f$, i.e., the domain of $f$ contains an open interval surrounding $x_0$. The derivative (also called first derivative) of $f$ at $x_0$, denoted $f^{\prime }(x_0)$ is defined as the limit of the difference quotient of $f$ between $x_0$ and $x$, as $x\to x_0$. Explicitly:

$f^{\prime }(x_0):=\underset{x\to x_0}{lim}\mathrm{\Delta }f(x,x_0)=\underset{x\to x_0}{lim}\frac{f(x)-f(x_0)}{x-x_0}$

If this limit exists, then we say that the derivative exists and has this value, and we say that the function is differentiable at the point. If the limit does not exist, then we say that the function is not differentiable at the point and the derivative does not exist.

Geometric definition

Suppose $f$ is a function and $x_0$ is a point in the interior of the domain of $f$, i.e., the domain of $f$ contains an open interval surrounding $x_0$. The derivative of $f$ at $x_0$ is the slope of the tangent line to the graph of $f$ through the point $(x_0,f(x_0))$.

Definition as a function

Suppose $f$ is a function defined on a subset of the reals. Its derivative or first derivative, denoted $f^{\prime }$, is a function defined as follows:

• The domain is the following subset of the domain of $f$: An element in the domain of $f$ is in the domain of $f^{\prime }$ if and only if it is in the interior of the domain of $f$ and the derivative of $f$ exists at the point.
• The function value at any point in the domain is simply the value of the derivative of $f$ at that point.

Leibniz notation for derivative

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