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Chapter 2: Differentiation
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Section 2.2: Precise Definition of the Derivative
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Introduction


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Section 2.1 introduced the idea of the derivative $f\prime$, explaining that it is a function giving the slope of the line tangent to the graph of $f$ at points in the domain of $f$. It was also interpreted as the instantaneous rate of change of $f$.
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This section will provide a formal definition of the derivative, and the next section will contain a useful list of differentiation rules.
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Essentials



The Derivative


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Definition 2.2.1

The derivative of the function f at $x$, denoted by the symbol $f\prime \left(x\right)$, is given by
$f\prime \left(x\right)equals;\underset{h\to 0}{lim}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}\frac{f\left(xplus;h\right)f\left(x\right)}{h}$
whenever the limit exists as a real (and therefore finite) number.
When the limit exists, f is said to be differentiable at $x$.
If this limit does not exist, then f is not differentiable at $x$.



Definition 2.2.1 is the formal statement that the derivative is the limiting slope of secant lines. To see this, write the slope of the secant line connecting the points$\left(x\,f\left(x\right)\right)$ and$\left(z\,f\left(z\right)\right)$ as
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$\frac{f\left(z\right)f\left(x\right)}{zx}$
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Then, set $h\=zx$ so that $z\=x\+h$. The slope of the secant line becomes
$\frac{f\left(x\+h\right)f\left(x\right)}{h}$
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Definition 2.2.2

The general expression for the slope of the secant line, namely,
$\frac{f\left(x\+h\right)f\left(x\right)}{h}$
is called the difference quotient. (In Maple, this is called the Newton quotient.)



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Notations for the Derivative


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There are several notations for the derivative of the function g. One of the most useful is $g\prime$, where, in Maple, the prime can be an apostrophe or the stroke found in the Punctuation palette. To evaluate a derivative at the specific point $x\=c$, use the prime notation $g\prime \left(c\right)$.
Another common notation is $\frac{\mathrm{df}}{\mathrm{dx}}$ which is implemented in Maple through the $\frac{\ⅆ}{\ⅆ\phantom{\rule[0.0ex]{0.2em}{0.0ex}}{x}}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}{f}$ template in the Calculus (or Expression) palette. This notation suffers from two defects: there is no simple analog of $g\prime \left(c\right)$; and in Maple, the spacing in the template can't be modified. Although the notation $\frac{\mathrm{dg}\left(c\right)}{\mathrm{dx}}$ does sometimes appear in the literature, taken literally it means $g\left(c\right)$ is evaluated first, then differentiated with respect to $x$, resulting in zero.
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These two seemingly different notations for the derivative can be traced to the historical origins of the calculus. In England, Isaac Newton developed his version of the calculus using terms like fluxions and fusions, and notation like $\stackrel{\.}{g}\left(t\right)$ for the derivative. In Germany, Gottfried Leibniz developed his version of the calculus using the differential notation $\frac{\mathrm{dy}}{\mathrm{dx}}$. Unfortunately, this notation makes it seem that the derivative is the ratio of finite quantities $\mathrm{dy}$ and $\mathrm{dx}$, called differentials. This is not the case, and it is only when the differential is defined as a derivative $y\prime$ times an increment $\mathrm{dx}$ that the notation of Leibniz becomes comfortable. Readers interested in the historical arguments as to primality of the development of the calculus can find ample material in any good mathematical library or on the internet. No reference will be given here because no need is seen to reopen what was a long and acrimonious debate as to who deserved credit for "inventing" the calculus.


Theorem 2.2.1


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Differentiability implies continuity, but continuity does not imply differentiability. Section 2.1 points out that at a cusp, corner, (where a function is continuous) it is not differentiable, so clearly, continuity does not imply differentiability. It is even possible for a function to be continuous everywhere, but differentiable nowhere!
The following theorem is true, but its converse is not.
Theorem 2.2.1

If a function $f$ is differentiable at $a$ then f is continuous at $a\.$



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Proof of Theorem 2.2.1

The differentiability of $f$ at $a$ means
$f\prime \left(a\right)\=\underset{h\to 0}{lim}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}\frac{f\left(a\+h\right)f\left(a\right)}{h}$
exists and is finite. To show f is continuous at $a$ , show that $\underset{x\to a}{lim}f\left(x\right)equals;f\left(a\right)$. For all $x\ne a$ it is valid to write
$f\left(x\right)$

$\=f\left(a\right)plus;f\left(x\right)f\left(a\right)$

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$\=$$f\left(a\right)\+\frac{f\left(x\right)f\left(a\right)}{xa}$ ($xa$ )



Applying the limit operator to both sides of this equation leads to
$\underset{x\to a}{lim}f\left(x\right)$

$\=\underset{x\to a}{lim}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}\left(f\left(a\right)\+\frac{f\left(x\right)f\left(a\right)}{xa}\left(xa\right)\right)$

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$\=f\left(a\right)\+\underset{x\to a}{lim}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}\left(\frac{f\left(x\right)f\left(a\right)}{xa}\right)\underset{x\to a}{lim}\left(xa\right)$



The limit of the difference quotient can be recognized as the derivative of $f$ at $a$, namely, $f\prime \left(a\right)$. Because this limit exists and is finite, the Product law for limits can be applied to conclude
$\underset{x\to a}{lim}f\left(x\right)$

$\=f\left(a\right)\+f\prime \left(a\right)\cdot 0$

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$\=f\left(a\right)$



Thus, f must be continuous at $a$.



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Précis


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The content of Definition 2.2.1 is summarized by
$\genfrac{}{}{0ex}{}{\frac{\mathrm{df}}{\mathrm{dx}}}{\phantom{x\=a}}\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=c}\=f\prime \left(c\right)\=$$\underset{h\to 0}{lim}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}\frac{f\left(c\+h\right)f\left(c\right)}{c}$ = $\underset{z\to c}{lim}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}\frac{f\left(z\right)f\left(c\right)}{zc}$
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provided the limit exists (and is finite). These symbols formalize the idea that the limiting slopes of secant lines is the slope of the tangent line, and that the derivative is essentially the slope of the tangent line.
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Examples


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Example 2.2.1

Apply Definition 2.2.1 to $f\left(x\right)\={x}^{3}$, to obtain $f\prime \left(c\right)$, the derivative at the point $\left(c\,f\left(c\right)\right)$.

Example 2.2.2

Apply Definition 2.2.1 to $f\left(x\right)\=\frac{{x}^{2}\+1}{x\+2}$, to obtain $f\prime \left(c\right)$, $c\ne 2$.

Example 2.2.3

Apply Definition 2.2.1 to $f\left(x\right)\=\sqrt{x1}$, to obtain $f\prime \left(c\right)$, $c\>1$.

Example 2.2.4

Evaluate $\underset{z\to 2}{lim}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}\frac{\sqrt{{z}^{2}\+1}\sqrt{5}}{z2}$.



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