Differentiable function
High school
Consider a function $f$ and a point $a$ in its domain. The function $f$ is differentiable at $a$ if the limit
\[f^{\prime}(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\]exists. It is differentiable if this holds at every point of its domain.
Undergraduate (functions on open subsets of $\reals^n$)
Consider a function $f$ on an open subset of $\reals^n$ and a point $a$ in its domain. The function $f$ is differentiable at $a$ if there is a linear map $Df(a)$ with
\[\lim_{h \to 0} \frac{\norm{f(a + h) - f(a) - Df(a) h}}{\norm{h}} = 0.\]Examples
Examples
-
$x \mapsto \sin x$
High school
$x \mapsto \sin x$ is differentiable at every $x$, with derivative $\cos x$.
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$x \mapsto x \abs{x}$
High school
$x \mapsto x \abs{x}$ is differentiable at every $x$: away from $0$, the function equals $x^2$ or $-x^2$, and at $0$, the difference quotient $\frac{h \abs{h}}{h} = \abs{h}$ tends to $0$, so the derivative at $0$ is $0$.
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$f(x) = x^2 \sin(1/x)$ for $x \neq 0$, $f(0) = 0$
Undergraduate
$f(x) = x^2 \sin(1/x)$ for $x \neq 0$, $f(0) = 0$, is differentiable at every $x$: away from $0$, the product and chain rules give $f^{\prime}(x) = 2x \sin(1/x) - \cos(1/x)$, and at $0$, the difference quotient $\frac{h^2 \sin(1/h)}{h} = h \sin(1/h)$ tends to $0$ because $\abs{h \sin(1/h)} \leq \abs{h}$, so $f^{\prime}(0) = 0$.
Non-examples
Things that may be mistaken for examples.
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$f(x) = \abs{x}$
High school
$x \mapsto \abs{x}$ is not differentiable at $0$, because it has a corner there.
Properties
- Every differentiable function is continuous.
- Having all partial derivatives at $a$ does not imply differentiability at $a$. If the partial derivatives exist near $a$ and are continuous at $a$, then $f$ is differentiable at $a$.
Distinctions involving this term
- Differentiable vs. continuously differentiable: For a function of a real variable, the derivative $f^{\prime}(x)$ measures the slope of the graph at each point $x$. A differentiable function has a derivative at every point, but the derivative can oscillate wildly as $x$ changes causing the derivative to be discontinuous. A continuously differentiable function has a derivative at every point, and the derivative is also a continuous function of $x$.