Differentiability

Section 1.1 Differentiability

The derivative of a function

f (x)

at a point

x_{0}

is the instantaneous rate of change of $f (x)$ with respect to $x$ at the point

x = x_{0} .

Geometrically, it is the slope of the tangent at the point $(x_{0}, f (x_{0}))$ of the graph

y = f (x) .

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Given a description of how a quantity

y

changes with respect to a quantity

x

as a function

y (x),

we often interested in how fast

y

is changing against

x .

The rate of change of

y

with respect to the change of

x

is easy to understand: it is simply the ratio of the change of

y

over the change of

x .

That is,

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\frac{Δ y}{Δ x} = \frac{y (x_{2}) - y (x_{1})}{x_{2} - x_{1}} .

For instance, if you traveled 140 miles in 2 hours then your average speed over that 2 hours period is

140 / 2 = 70

miles per hour (mph). But, as you can imagine, an average speed of

70

mph, does not mean you traveled constantly as speed

70

mph during that 2 hour period.

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Things indeed get trickier when we are interested in the instantaneous rate of change of

y

with respect to

x .

In view of the example above, that is, we are interested in the speedometer reading of your car. Suppose the quantity

y

is given by a function

f

x

and we are interested in the instantaneous rate of change of

y

with respect to

x

when

x

is at the value

x_{0} .

As we have discussed, for

x \neq x_{0}

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φ_{x_{0}} (x) = \frac{f (x) - f (x_{0})}{x - x_{0}}

is the rate of change of

y

with respect to

x .

However, we cannot simply evaluate

φ_{x_{0}} (x)

x = x_{0}

to obtain the instantaneous rate of change as the denominator vanishes when

x = x_{0} .

And yet, we do expect

φ_{x_{0}} (x)

is close to the instantaneous rate of change that we are looking for when

x

is close to

x_{0} .

Hence, it reasonable to say that if

φ_{x_{0}} (x)

can be extended to a function that is continuous at

x = x_{0},

then the value of this extension at

x = x_{0}

is the instantaneous rate of change of

y

x = x_{0} .

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Definition 1.1.

A function

f (x)

is differentiable at

x = x_{0}

if there exists a unique function

φ_{x_{0}} (x)

continuous at $x = x_{0}$ such that for all

x

at which

f

is defined,

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\begin{matrix} (1.1) & f (x) - f (x_{0}) = φ_{x_{0}} (x) (x - x_{0}) . \end{matrix}

The value

φ_{x_{0}} (x_{0}),

also denoted by

f^{'} (x_{0}),

is called the derivative of

f (x)

with respect to

x

x = x_{0} .

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Remark 1.2.

This definition of derivative is due to Constantin Caratheodory.
We are simply defining differentiability in terms of continuity. But we hope continuity is a more intuitive concept and is easier to understand.
For the expert: it is implicit in the definition that $f$ is defined at $x_{0} .$ Moreover, the uniqueness of the extension of $φ_{x_{0}} (x)$ is equivalent to $x_{0}$ being a limit point of the domain of $f .$ Note also that the uniqueness of the extension is automatic when the domain of $f$ is an interval (with at least two distinct points).
Note also that differentiability is a local property. That means if $f$ and $g$ agree on an open interval containing $a,$ then $f^{'} (a) = g^{'} (a)$ if either of them exists.

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Geometrically

φ_{x_{0}} (x)

is the slope of the secant line joining the point

(x_{0}, f (x_{0}))

and

(x, f (x)) .

So, if

f (x)

is differentiable at

x = x_{0},

then as

x

approaches

x_{0}

these secants approach the line defined by

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\begin{matrix} (1.2) & f (x_{0}) + f^{'} (x_{0}) (x - x_{0}) \end{matrix}

which is called the tangent of

f (x)

x = x_{0} .

It is the line that passes through the point

(x_{0}, f (x_{0}))

with slope

f^{'} (x_{0}) .

In other words

f^{'} (x_{0}),

if exists, is the slope of the tangent of the graph

y = f (x)

at the point

(x_{0}, f (x_{0})) .

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Using

f^{'} (x_{0})

to denote the derivative of

f (x)

x = x_{0}

is due to Lagrange. This notation suggests that the derivatives of

f (x)

at various points are the values of another function namely

f^{'} (x) .

Another popular notation for the derivative of

f (x)

x = x_{0}

\frac{d f}{d x} (x_{0}) .

This notation is due to Leibniz. It reminds us that the derivative is the rate of change of

f (x)

with respect to

x

x_{0} .

It emphasizes the variable is

x .

If we do not a specific name of the function but simply consider its values as a dependent valuable

y

depending on

x,

then we also write its derivative at

x = x_{0}

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\frac{d y}{d x} (x_{0}), {\frac{d y}{d x} |}_{x = x_{0}}, or {\frac{d y}{d x} |}_{(x_{0}, f (x_{0}))} .

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Example 1.3.

For a linear function

f (x) = m x + b,

it is plain to verify that

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f (x) = f (x_{0}) + m (x - x_{0})

for any

x, x_{0} .

Constant functions are continuous so we have

f^{'} (x_{0}) = m

for any

x_{0} .

In other words, the derivative of

f (x) = m x + b

(as a function of

x

) is the constant function

f^{'} (x) = m .

In particular, the derivative of a constant function

f (x) = b

is the zero function. And the derivative of the identity function

f (x) = x

is the constant function

1 .

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Example 1.4.

Consider the function

f (x) = 1 / x .

For

x, x_{0} \neq 0,

since

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f (x) - f (x_{0}) = \frac{1}{x} - \frac{1}{x_{0}} = - \frac{1}{x x_{0}} (x - x_{0})

and the function

φ_{x_{0}} (x) = - 1 / (x x_{0})

is continuous at

x_{0},

we conclude that

f^{'} (x_{0}) = - 1 / x_{0}^{2} .

And so the derivative of

1 / x

is the function

- 1 / x^{2} .

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Proposition 1.5.

f

is differentiable at

x_{0}

then

f

is continuous at

x_{0} .

In particular, differentiable functions are continuous.

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Proof.

By definition of differentiability,

f (x) = f (x_{0}) + φ (x) (x - x_{0})

for some function

φ (x)

that is continuous at

x = x_{0} .

Therefore,

f

is continuous at

x_{0}

since constant functions are continuous and continuous functions are closed under sum and products.

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