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Chapter 2 Derivatives

“...the secret of education lies in respecting the pupil...” - Ralph Waldo Emerson

Up to this point, we have solved several different problems and discovered that the same concept has generated a solution each time. Limits were used to find:

  • Robbie's velocity function given his position function,

  • the instantaneous growth function for the robot population given the population function, and

  • the function that tells us the slope of the line tangent to a function at any point.

The following definition is the result of all of our work to this point.

Definition 2.1.

Given a function \(f\text{,}\) the derivative of \(f\text{,}\) denoted by \(f'\text{,}\) is the function so that for every point \(x\) in the domain of \(f\) where \(f\) has a tangent line, \(f'(x)\) equals the slope of the tangent line to \(f\) at \((x,f(x)).\) Either

\begin{equation*} f'(x) = \lim_{b \rightarrow x} \frac{f(b)-f(x)}{b-x} \; \; \; \; \mbox{ or } \; \; \; \; f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \end{equation*}

may be used to compute the derivative.

While both limits above will always yield the same result, there are times when one may be preferable to the other in terms of the algebraic computations required to obtain the result. Given a function \(f\text{,}\) the derivative of \(f\) is usually denoted by \(\dsp f', \;\; \frac{df}{dx} \;\; \mbox{or} \;\; \dot{f}.\) Regardless of how it is written, the derivative of \(f\) at \(x\) represents the slope of the line tangent to \(f\) at the point \((x,f(x))\) or the rate at which the quantity \(f\) is changing at \(x\text{.}\) The origins of calculus are generally credited to Leibnitz and Newton. The first notation, credited to Lagrange, is referred to as prime or functional notation. The second notation is due to Leibnitz. The third is due to Newton. Engineers sometimes use the dot to represent derivatives that are taken with respect to time.

If \(f(x) = x^2,\) then each of

\begin{equation*} f'(x) \mbox{ and } ( x^2 )' \mbox{ and } \frac{d}{dx} (x^2) \end{equation*}

all mean “the derivative of \(f\) at \(x\text{.}\)”

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