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Section2.2Definition of the derivative

We now define the “derivative” explicitly, based on the limiting slope ideas of the previous section. Then we see how to compute some simple derivatives.

Let us now generalise what we did in the last section so as to find “the slope of the curve \(y=f(x)\) at \((x_0,y_0)\)” for any smooth enough  1  function \(f(x)\text{.}\)

As before, let \((x_0,y_0)\) be any point on the curve \(y=f(x)\text{.}\) So we must have \(y_0=f(x_0)\text{.}\) Now let \((x_1,y_1)\) be any other point on the same curve. So \(y_1=f(x_1)\) and \(x_1\ne x_0\text{.}\) Think of \((x_1,y_1)\) as being pretty close to \((x_0,y_0)\) so that the difference

\begin{gather*} \De x=x_1-x_0 \end{gather*}

in \(x\)–coordinates is pretty small. In terms of this \(\De x\) we have

\begin{gather*} x_1=x_0+\De x\qquad\text{and}\qquad y_1=f\big(x_0+\De x\big) \end{gather*}

We can construct a secant line through \((x_0,y_0)\) and \((x_1,y_1)\) just as we did for the parabola above. It has slope

\begin{gather*} \frac{y_1-y_0}{x_1-x_0}=\frac{f\big(x_0+\De x\big)-f(x_0)}{\De x} \end{gather*}

If \(f(x)\) is reasonably smooth  2  , then as \(x_1\) approaches \(x_0\text{,}\) i.e. as \(\De x\) approaches \(0\text{,}\) we would expect the secant through \((x_0,y_0)\) and \((x_1,y_1)\) to approach the tangent line to the curve \(y=f(x)\) at \((x_0,y_0)\text{,}\) just as happened in Figure 2.1.6. And more importantly, the slope of the secant through \((x_0,y_0)\) and \((x_1,y_1)\) should approach the slope of the tangent line to the curve \(y=f(x)\) at \((x_0,y_0)\text{.}\)

Thus we would expect  3 

the slope of the tangent line to the curve \(y=f(x)\) at \((x_0,y_0)\) to be

\begin{gather*} \lim_{\De x\rightarrow 0}\frac{f\big(x_0+\De x\big)-f(x_0)}{\De x} \end{gather*}

When we talk of the “slope of the curve” at a point, what we really mean is the slope of the tangent line to the curve at that point. So “the slope of the curve \(y=f(x)\) at \((x_0,y_0)\)” is also the limit  4 

expressed in the above equation. The derivative of \(f(x)\) at \(x=x_0\) is also defined to be this limit. Which leads  5  us to the most important definition in this text:

Definition2.2.1Derivative at a point

Let \(a\in\bbbr\) and let \(f(x)\) be defined on an open interval  6 

that contains \(a\text{.}\)

  • The derivative of \(f(x)\) at \(x=a\) is denoted \(f'(a)\) and is defined by \begin{gather*} f'(a)=\lim_{h\rightarrow 0}\frac{f\big(a+h\big)-f(a)}{h} \end{gather*} if the limit exists.
  • When the above limit exists, the function \(f(x)\) is said to be differentiable at \(x=a\text{.}\) When the limit does not exist, the function \(f(x)\) is said to be not differentiable at \(x=a\text{.}\)
  • We can equivalently define the derivative \(f'(a)\) by the limit \begin{gather*} f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}. \end{gather*} To see that these two definitions are the same, we set \(x=a+h\) and then the limit as \(h\) goes to \(0\) is equivalent to the limit as \(x\) goes to \(a\text{.}\)

Lets now compute the derivatives of some very simple functions. This is our first step towards building up a toolbox for computing derivatives of complicated functions — this process will very much parallel what we did in Chapter 1 with limits. The two simplest functions we know are \(f(x)=c\) and \(g(x)=x\text{.}\)

That was easy! What about the next most complicated function — arguably it's this one:

That was a little harder than the first example, but still quite straight forward — start with the definition and apply what we know about limits.

Thanks to these two examples, we have our first theorem about derivatives:

To ratchet up the difficulty a little bit more, let us redo the example we have already done a few times \(f(x)=x^2\text{.}\) To make it a little more interesting let's change the names of the function and the variable so that it is not exactly the same as Examples  2.1.2 and 2.1.5.