Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 22. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | The Derivative Function 191 Where f x 0 the graph of f has a horizontal tangent line. 2x 3 0 2x -3 x Below we graph f and f . Notice that the answers make sense in terms of our intuitive ideas about slope. EXERCISE Use the limit definition of derivative to find the derivatives of x2 and 3x. Conclude that the derivative of x2 3x is the sum of the derivatives of x2 and 3x. Notice that so far the strategy that has been working for computing derivatives is to simplify the difference quotient to the point that we can factor an h from the numerator to cancel with the h in the denominator assuming that h 0 and once done the limit becomes apparent. Sometimes the simplification allowing cancellation is a bit more complicated as illustrated in the next problem. You should compute some simpler derivatives on your own before reading the next example. It is designed for your second reading of the chapter. EXAMPLE Let f x V . Find f x . SOLUTION f x h - f x f x lim ------------- h 0 h Vx h a x lim --------- h 0 h We have a dilemma here to make progress we must cancel the h in the denominator. This means getting rid of the square roots. We re working with an expression so our options are to multiply by 1 multiply numerator and denominator by the same nonzero quantity or to add zero. The former will be most productive. Multiplying the expression Va - VB by Va VB will eliminate the square Va - Vb is called the conjugate of VA VB . Va - Vb 7a VB a - Tab Tab - b a - b 8Multiplying VA VB by itself does not eliminatethe square roots. VA VB VA VB A 2VAB B which leaves the square root in the mixed term. 192 CHAPTERS The Derivative Function This algebraic maneuver is worth stashing away in your mind for future reference. . y A - - h a A j----- . f x lim -----------Multiply numerator and denominator byvx h s x. h 0 h . Ux h x wx h 4x lim ------------ - ------------ h o h Vx h s x h lim --- ------ h o h Vx h s x lim ---- ---- Since h 0 - 1. h o h x h x h 1 lim .
