Section 2 7 The Derivative as a Function Differentiable A function f x is differentiable at x a means f a exists If the derivative exists on an interval that is if f is differentiable at every point in the interval then the derivative is a function on that interval f x h f x f x lim h h 0 Definition Example f x x 2 f x lim x h h 0 2 x 2 lim x 2 2 xh h h 0 h h 2 x 2 lim 2 x h 2 x h 0 We also know the derivative of a line is the slope of the line Applying the linear rule for derivatives we can differentiate any quadratic Example g x 4 x 2 6 x 7 g x 4 2 x 6 8 x 6 We can find the tangent line to g at a given point Example For g x as above find the equation of the tangent line at 2 g 2 g 2 4 4 12 7 11 so the point of tangency is 2 11 The slope is g 2 8 2 6 10 so the tangent line is y 10 x 2 11 Notations for the derivative f x df f a dx df dx x a If a function is differentiable at x a then it must be continuous at a Contrapositive If a function is not continuous at x a then it is not differentiable at a The following example illustrates why Example 2 x f x 5 x 2 x 2 This is not continuous at 2 so why does that mean it is not differentiable at 2 lim h 0 f 2 h f 2 h lim h 0 2 h 2 5 As h approaches 0 the numerator approaches 1 h and the denominator approaches 0 the left side approaches infinity and the right side approaches minus infinity and the limit does not exist Graphically you cannot draw a line tangent to the graph at x 2 and passing through 2 5 A function is not differentiable where it has a corner a cusp a vertical tangent or at any discontinuity These are some possibilities we will cover Examples of corners and cusps 1 f x x 2 4 This function turns sharply at 2 and at 2 It is not differentiable at x 2 or at x 2 To graph it sketch the graph of negative across the x axis 2 f x x 2 3 x 2 4 and reflect the region where y is Graph it in your calculator and you will see the cusp at 0 0 3 When a piecewise function is continuous at a but the left and right pieces meet at different slopes the function has a corner at a x f x 5 x 6 2 Example x 3 3 x Check that f is continuous at 3 The slope of the left piece at 3 is the derivative of x 2 at x 3 We found the derivative of 2 x is 2x so the slope of the left piece at 3 is 6 The slope of the right piece is 5 The pieces meet at an obtuse corner A more obvious corner occurs in x at x 0 Vertical Tangents occur when f is continuous but f has a vertical asymptote 1 3 has a vertical tangent at x 0 g x x Example f x x 2 4 1 3 Graph this function in your calculator It has vertical tangents at x 2 and at x 2 We will learn how to find this derivative in later sections Higher Order Derivatives Since f is a function we can try to find its derivative The derivative of f is the 2nd derivative of f and written f x d 2 dx Example f 2 f x x 2 f x 2 x f x 2 Velocity is the derivative of distance traveled and acceleration is the derivative of velocity so acceleration is the 2nd derivative of distance Example The height of an object shot straight upward from 60 ft above the ground with initial velocity 50 ft per sec has height given by 2 h t 50 60 t 16 t The velocity at t sec is h t v t 60 32 t ft sec and the acceleration is gravity h t v t 32 ft sec 2 which is the acceleration downward due to