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# DIFFERENTIAL CALCULUS MTH 251H

OSU

GPA 3.79

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This 9 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 251H at Oregon State University taught by Staff in Fall. Since its upload, it has received 38 views. For similar materials see /class/224448/mth-251h-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15

Section 29 The Derivative as a Function In the de nition of the derivative of a function f at the point a h u hmfafL fa hgt0 we regarded the point a as xed Now we set a free De nition Let fac be a function of ii The derivative of f is the function denoted by f 7 whose domain consists of all x in the domain of the function f for which 1 fxh fx 1m hgt0 h exists and whose value at such an x is The function f is derived from the function f by the limiting process indicated above Thus the name derivative for f In geometric terms f the slope of the tangent line to the graph of f at the point xf Terminology and Notation We say a function f is di ei entiable on an interval I if f exists for all so in I with appropriate one sided derivatives understood at the endpoints of I that belong to I If y f is a function of x then the following notations are all used to denote the derivative of f at x df 7 gm we we I I N90 3 dx dx The fractional forms stem from Leibniz as a suggestive way to remember that derivatives stem from limits of difference quotients dy Av i 1m 7 dac A140 A90 The symbols D and D7 are shorthand instructions that mean take the deriv ative of the function that follows The prime notation f and y are due to Lagrange and came well after the Leibniz notation Newton often denoted a derivatives by a dot placed above an expression Today this notation is used primarily in connection with motion problems Differentiation Rules I Secs 31 amp 32 The differentiation rules of calculus enable you to differentiate virtually any function that comes up in practical applications as soon as you know how to differentiate a few basic functions 7 constant functions power functions the six trigonometric functions exponential functions and logarithmic functions There are seven fundamental rules of differentiation and some variations on a theme They are the quot divide and conquerquot rules of differentiation The most basic ve are 1 f x g a if x gm 2 M 7m gm 7 gm 3 gen cfx 4 gm MWfxggxfr 5 g 9xf91xgx These rules are valid whenever the right members are de ned When is that The laws one and two are called the sum and difference rules and laws four and ve are the product rule and the quotient rule Notice that the second law is a consequence of laws one and three Why A very useful consequence of laws one and three is d d d 7 a or b or a7 or b7 or dxlt f gt glt gtgt d gt dxgm for any constants a and b This result includes both laws one two and three How can you recover laws one two and three from the displayed rule Any process like differentiation that has properties 1 and 3 or equivalently the displayed property is called a linear process or linear operation or linear operator So differentiation is a linear operator Next term you will learn that integration whatever that is is a linear operator Since 1 stands for any constant whatsoever in the foregoing equality you can replace 1 by the constant 71 and obtain d af x 7 by an agf x 7 bag as You must know how to differentiate something to make the foregoing rules useful Here are rst steps in adding value to the algebraic rules for differentiation A Catalog of Basic Derivatives d D1 70 0 for any constant 0 do D2 790quot nacquot 1 for any integer n do D3 dixr rxT l for any real number 7 and x gt 0 90 d D4 7e e for any real 90 do 1 D5 flux7 for allxgt0 div ac D6 Ea 1n 1 a for any real ac d 1 D7 71 7 f 11 gt 0 dgc ogax Ina 90 or a 90 D2 is called the power rule and D3 is the general power rule D4 is a special case of D6 and D5 is a special case of D7 Why Without a single calculation we can make a simple observation based on D1 D2 the linearity of differentiation and the quotient rule All polynomials and all rational functions are differentiable In particular cases the derivative of a polynomial can be written down by inspec tion and the derivative of a rational function can be obtained without much more work using the quotient rule Which of the graphs represents the position of an object that is slowing down a distance b distance 3 met 1 dislanee 4 Which of the following tables could represent an exponential function 1111 1 1111 l 1 111 1111 1 111 1 9 1 1 2 42 2 4 2 s 2 1 4 1s 3 z 4 413 a 114 4 1 The graph in Figure 110 is that of y fx Which of the graphs 1 IV could be a graph of cfx Flgule 1n mu m The graph in Figure 110 is that of y fx Which of the graphs 1 IV could be a graph of fx k Flgule 1n mu m The graph in Figure 110 is that of y fx Which of the graphs 1 IV could be a graph of fx h Flgule 1n mu m

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