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Subsection 2.2.1 An Important Point (and Some Notation)

Notice here that the answer we get depends on our choice of \(a\) — if we want to know the derivative at \(a=3\) we can just substitute \(a=3\) into our answer \(2a\) to get the slope is 6. If we want to know at \(a=1\) (like at the end of Section 1.1) we substitute \(a=1\) and get the slope is 2. The important thing here is that we can move from the derivative being computed at a specific point to the derivative being a function itself — input any value of \(a\) and it returns the slope of the tangent line to the curve at the point \(x=a\text{,}\) \(y=h(a)\text{.}\) The variable \(a\) is a dummy variable. We can rename \(a\) to anything we want, like \(x\text{,}\) for example. So we can replace every \(a\) in

\begin{align*} h'(a)&=2a &\text{ by $x$, giving} && h'(x) &=2x \end{align*}

where all we have done is replaced the symbol \(a\) by the symbol \(x\text{.}\)

We can do this more generally and tweak the derivative at a specific point \(a\) to obtain the derivative as a function of \(x\text{.}\) We replace

\begin{align*} f'(a) &= \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\\ \end{align*}

with

\begin{align*} f'(x) &= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \end{align*}

which gives us the following definition

Definition 2.2.6 Derivative as a function

Let \(f(x)\) be a function.

  • The derivative of \(f(x)\) with respect to \(x\) is
    \begin{gather*} f'(x)=\lim_{h\rightarrow 0}\frac{f\big(x+h\big)-f(x)}{h} \end{gather*}
    provided the limit exists.
  • If the derivative \(f'(x)\) exists for all \(x \in (a,b)\) we say that \(f\) is differentiable on \((a,b)\text{.}\)
  • Note that we will sometimes be a little sloppy with our discussions and simply write “\(f\) is differentiable” to mean “\(f\) is differentiable on an interval we are interested in” or “\(f\) is differentiable everywhere”.

Notice that we are no longer thinking of tangent lines, rather this is an operation we can do on a function. For example:

Let \(f(x) = \frac{1}{x}\) and compute its derivative with respect to \(x\) — think carefully about where the derivative exists.

  • Our first step is to write down the definition of the derivative — at this stage, we know of no other strategy for computing derivatives.
    \begin{align*} f'(x)&=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} && \text{(the definition)} \end{align*}
  • And now we substitute in the function and compute the limit.
    \begin{align*} f'(x)&=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} && \text{(the definition)}\\ &=\lim_{h\rightarrow 0}\frac{1}{h}\left[\frac{1}{x+h}-\frac{1}{x}\right] && \text{(substituted in the function)}\\ &=\lim_{h\rightarrow 0}\frac{1}{h}\ \frac{x-(x+h)}{x(x+h)} && \text{(wrote over a common denominator)}\\ &=\lim_{h\rightarrow 0}\frac{1}{h}\ \frac{-h}{x(x+h)} && \text{(started cleanup)}\\ &=\lim_{h\rightarrow 0} \frac{-1}{x(x+h)}\\ &=-\frac{1}{x^2} \end{align*}
  • Notice that the original function \(f(x)=\frac{1}{x}\) was not defined at \(x=0\) and the derivative is also not defined at \(x=0\text{.}\) This does happen more generally — if \(f(x)\) is not defined at a particular point \(x=a\text{,}\) then the derivative will not exist at that point either.

So we now have two slightly different ideas of derivatives:

  • The derivative \(f'(a)\) at a specific point \(x=a\text{,}\) being the slope of the tangent line to the curve at \(x=a\text{,}\) and
  • The derivative as a function, \(f'(x)\) as defined in Definition 2.2.6.

Of course, if we have \(f'(x)\) then we can always recover the derivative at a specific point by substituting \(x=a\text{.}\)

As we noted at the beginning of the chapter, the derivative was discovered independently by Newton and Leibniz in the late \(17^{\rm th}\) century. Because their discoveries were independent, Newton and Leibniz did not have exactly the same notation. Stemming from this, and from the many different contexts in which derivatives are used, there are quite a few alternate notations for the derivative:

Definition 2.2.8

The following notations are all used for “the derivative of \(f(x)\) with respect to \(x\)”

\begin{gather*} f'(x) \qquad \diff{f}{x} \qquad \diff{}{x}f(x) \qquad \dot{f}(x) \qquad Df(x) \qquad D_x f(x), \end{gather*}

while the following notations are all used for “the derivative of \(f(x)\) at \(x=a\)”

\begin{gather*} f'(a) \qquad \diff{f}{x}(a) \qquad \diff{}{x}f(x)\,\bigg|_{x=a} \qquad \dot{f}(a) \qquad Df(a) \qquad D_x f(a). \end{gather*}

Some things to note about these notations:

  • We will generally use the first three, but you should recognise them all. The notation \(f'(a)\) is due to Lagrange, while the notation \(\diff{f}{x}(a)\) is due to Leibniz. They are both very useful. Neither can be considered “better”.
  • Leibniz notation writes the derivative as a “fraction” — however it is definitely not a fraction and should not be thought of in that way. It is just shorthand, which is read as “the derivative of \(f\) with respect to \(x\)”.
  • You read \(f'(x)\) as “\(f\)–prime of \(x\)”, and \(\diff{f}{x}\) as “dee–\(f\)–dee–\(x\)”, and \(\diff{ }{x}f(x)\) as “dee-by-dee–\(x\) of \(f\)”.
  • Similarly you read \(\diff{f}{x}(a)\) as “dee–\(f\)–dee–\(x\) at \(a\)”, and \(\diff{ }{x}f(x)|_{x=a}\) as “dee-by-dee-\(x\) of \(f\) of \(x\) at \(x\) equals \(a\)”.
  • The notation \(\dot f\) is due to Newton. In physics, it is common to use \(\dot f(t)\) to denote the derivative of \(f\) with respect to time.
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