Differentiable
Differentiable
Differentiable means that the
exists ...
Example: is x2 + 6x differentiable?
tell us the derivative of x2 is 2x and the derivative of x is 1, so:
Its derivative is 2x + 6
So yes! x2 + 6x is differentiable.
... and it must exist for every value in the function's .
In its simplest form the domain is
all the values that go into a function
Example (continued)
When not stated we assume that the domain is the .
For x2 + 6x, its
derivative of 2x + 6 exists for all Real Numbers.
So we are still safe: x2 + 6x is differentiable.
But what about this:
Example: The function f(x) = |x| ():
|x| looks like this:
At x=0 it has a very pointy change!
Does the derivative exist
We can test any value &c& by finding if the
f(c+h) & f(c)
Example (continued)
Let's calculate the limit for |x| at the value 0:
|0+h| & |0|
The limit does not exist
To see why, let's compare
and right side limits:
From Left Side:
From Right Side:
The limits are different on either side, so the limit does not exist.
So the function f(x) = |x| is not differentiable
A good way to picture this in your mind is to think:
As I zoom in, does the function tend to become a straight line?
The absolute value function stays
pointy even when zoomed in.
Other Reasons
Here are a few more examples:
are not differentiable at integer values, as there is a discontinuity at each jump. But they are differentiable elsewhere.
The Cube root function x(1/3)
Its derivative is (1/3)x-(2/3) (by the )
At x=0 the derivative is undefined, so
x(1/3) is not differentiable.
At x=0 the function is not defined so it makes no sense to ask if they are differentiable there.
To be differentiable at a certain point, the function must first of all be defined there!
As we head towards x = 0 the function
up and down faster and faster, so we cannot find a value it is &heading towards&.
So it is not differentiable.
Different Domain
But we can change the domain!
Example: The function g(x) = |x| with Domain (0,+&)
The domain is from but not including 0 onwards (all positive values).
Which IS differentiable.
And I am &absolutely positive& about that :)
So the function g(x) = |x| with Domain (0,+&) is differentiable.
We could also restrict the domain in other ways to avoid x=0 (such as all negative Real Numbers, all non-zero Real Numbers, etc).
Why Bother?
Because when a function is differentiable we can use all the power of calculus when working with it.
Continuous
When a function is differentiable it is also .
Differentiable &rA Continuous
But a function can be continuous but not differentiable. For example the absolute value function is actually continuous (though not differentiable) at x=0.
Copyright &From Wikipedia, the free encyclopedia
A differentiable function
(a branch of ), a differentiable function of one
variable is a function whose
exists at each point in its . As a result, the
of a differentiable function must have a (non-)
at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or .
More generally, if x0 is a point in the domain of a function f, then f is said to be differentiable at x0 if the derivative f ′(x0) exists. This means that the graph of f has a non-vertical tangent line at the point (x0, f(x0)). The function f may also be called locally linear at x0, as it can be well approximated by a
near this point.
function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis.
An ordinary
on the cubic curve () x3 – y2 = 0, which is equivalent to the
f(x) = ± x3/2. This relation is continuous, but is not differentiable at the cusp.
If f is differentiable at a point x0, then f must also be
at x0. In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, , or
may be continuous, but fails to be differentiable at the location of the anomaly.
Most functions that occur in practice have derivatives at all points or at
point. However, a result of
states that the set of functions that have a derivative at some point is a
in the space of all continuous functions. Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the .
Main article:
Differentiable functions can be locally approximated by linear functions.
The function
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
{\displaystyle f(x)=x^{2}\sin \left({\tfrac {1}{x}}\right)}
{\displaystyle x\neq 0}
{\displaystyle f(0)=0}
is differentiable. However, this function is not continuously differentiable.
A function f is said to be continuously differentiable if the derivative f'(x) exists and is itself a continuous function. Though the derivative of a differentiable function never has a , it is possible for the derivative to have an essential discontinuity. For example, the function
{\displaystyle f(x)\;=\;{\begin{cases}x^{2}\sin(1/x)&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}}
is differentiable at 0, since
{\displaystyle f'(0)=\lim _{\epsilon \to 0}\left({\frac {\epsilon ^{2}\sin(1/\epsilon )-0}{\epsilon }}\right)=0,}
exists. However, for x≠0,
{\displaystyle f'(x)=2x\sin(1/x)-\cos(1/x)}
which has no limit as x → 0. Nevertheless,
implies that the derivative of any function satisfies the conclusion of the .
Sometimes continuously differentiable functions are said to be of class C1. A function is of class C2 if the first and
of the function both exist and are continuous. More generally, a function is said to be of class Ck if the first k derivatives f′(x), f″(x), ..., f(k)(x) all exist and are continuous. If derivatives f(n) exist for all positive integers n, the function is
or equivalently, of class C∞.
If all the partial derivatives of a function exist and are continuous in a
of a point, then the function is differentiable at that point, and it is of class C1.
Formally, a
f: Rm → Rn is said to be differentiable at a point x0 if
J: Rm → Rn such that
{\displaystyle \lim _{\mathbf {h} \to \mathbf {0} }{\frac {\|\mathbf {f} (\mathbf {x_{0}} +\mathbf {h} )-\mathbf {f} (\mathbf {x_{0}} )-\mathbf {J} \mathbf {(h)} \|_{\mathbf {R} ^{n}}}{\|\mathbf {h} \|_{\mathbf {R} ^{m}}}}=0.}
If a function is differentiable at x0, then all of the
exist at x0, in which case the linear map J is given by the . A similar formulation of the higher-dimensional derivative is provided by the
found in single-variable calculus.
Note that existence of the partial derivatives (or even all of the ) does not in general guarantee that a function is differentiable at a point. For example, the function f: R2 → R defined by
{\displaystyle f(x,y)={\begin{cases}x&{\text{if }}y\neq x^{2}\\0&{\text{if }}y=x^{2}\end{cases}}}
is not differentiable at (0, 0), but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function
{\displaystyle f(x,y)={\begin{cases}y^{3}/(x^{2}+y^{2})&{\text{if }}(x,y)\neq (0,0)\\0&{\text{if }}(x,y)=(0,0)\end{cases}}}
is not differentiable at (0, 0), but again all of the partial derivatives and directional derivatives exist.
Main article:
In , any function that is complex-differentiable in a neighborhood of a point is called . Such a function is necessarily infinitely differentiable, and in fact .
If M is a , a real or complex-valued function f on M is said to be differentiable at a point p if it is differentiable with respect to some (or any) coordinate chart defined around p. More generally, if M and N are differentiable manifolds, a function f: M → N is said to be differentiable at a point p if it is differentiable with respect to some (or any) coordinate charts defined around p and f(p).
Banach, S. (1931). "?ber die Baire'sche Kategorie gewisser Funktionenmengen".
3 (1): 174–179.. Cited by Hewitt, E; Stromberg, K (1963). Real and abstract analysis. Springer-Verlag. Theorem 17.8.