Calculus 220 section 1 7 More about Derivatives notes by Tim Pilachowski Section 1 7 adds some depth to our discussion of derivatives of functions To review what we already have the following statements are mathematically equivalent a Find the slope of the line tangent to the graph of f at a point x y b Find lim h 0 f x h f x h c Find the first derivative of f x d Find f x e Find dy dx Recall however that the first derivative is itself a function which has its own domain and its own graph Since dy it is a function it also has its own derivative Given a function f we can calculate the first derivative f or dx d2y We can then calculate the derivative of f also called the second derivative of f symbolically f or dx 2 d2y dy Important note Just like is not a fraction but is a notation for the first derivative is also not a dx dx 2 fraction but a notation There is no multiplication involved Rather you need to interpret it this way d 2 y d dy dy the derivative of a derivative which means the derivative of 2 dx dx dx dx For now we ll focus on finding the first and second derivatives In later sections we ll be exploring what these two tell us about the original function f Example A Given f x x 3 8 x 2 find the first and second derivatives Answers 3 x 2 8 6x 2 Example B Given y 5 x 4 1 find dy d2y and Answers 200 x 7 40 x 3 1400 x 6 120 x 2 2 dx dx We can of course evaluate a function a first derivative or a second derivative at a specific point Example A revisited Given f x x 3 8 x 2 find f 1 f 1 and f 1 The point 1 9 is on the graph of f At the point 1 9 the graph of f has a slope of 5 At the point 1 9 f 6 and the curve is concave down 2 Example B revisited Given y 5 x 4 1 find y when x 1 dy dx and d2y dx 2 x 1 x 1 With this example a new notation is introduced for evaluating a derivative at a specific point At the point 1 16 the graph has a slope of 160 At the point 1 16 f 1280 and the curve is concave up In summary looking ahead to chapter 2 To find a point on the curve use the formula for y f x To find the slope of the curve use the formula for y f x dy dx To find the concavity of the curve use the formula for y f x d2y dx 2 It is possible to find not only first and second derivatives but also third fourth fifth etc Note that in the case of a polynomial all higher order derivatives would equal 0 specifically beginning with the n 1 st derivative of an nth degree polynomial Although in algebra we have traditionally thanks to Descartes used letters from the beginning of the alphabet a b and c and letters from the end of the alphabet x y and z to denote variables it does not have to be so In specific applications the letter is often chosen for ease of referring to what it represents The notations for derivatives are adjusted accordingly Example C The function s t 16t 2 v0 t s0 calculates the height of an object s after time t thrown with an initial velocity v0 and initial height s0 Write a function for a rock thrown upward at 10 feet per second from a bridge which is 240 feet above the river below Then find first and second derivatives s t and s t Answers 32t 10 32 Foreshadowing The first derivative in this situation describes velocity in feet per second and the second derivative describes acceleration in feet per second per second In business and economics you will often see p used to represent price x to represent demand Why not use d C to represent cost R to represent revenue and P to represent profit In this case the notation for derivatives would be adjusted to fit the situation Example D A small lampshade manufacturer has determined several equations which describe the economics 1 of his company The cost to produce x lampshades is C x x 2 4 x 200 Lampshades are sold for 8 p x 79 x Revenue is thus R x x p x x 79 x 79 x x 2 Write an equation to describe profit and find the first and second derivatives 2 d P x and d 2 P x Answers 9 x 2 75 x 200 9 x 75 9 dx 8 4 4 dx As a point of interest which we ll deal with in more depth in a later section C x is called the marginal cost R x is called the marginal revenue and P x is called the marginal profit The maximum profit will be found where P x R x C x 0 that is where R x C x