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# Download Derivatives of Constant and Linear Functions

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These notes closely follow the presentation of the material given in James Stewart’s textbook Calculus, Concepts and Contexts (2nd edition). These notes are intended primarily for in-class presentation and should not be regarded as a substitute for thoroughly reading the textbook itself and working through the exercises therein. Derivatives of Constant and Linear Functions If c is a constant, then the derivative of the function fx c (with domain all real numbers) is f x 0. In other words, d c 0. dx Example Let f be the function fx 13 (with domain all real numbers). Find the derivative of this function. If m and b are constants, then the derivative of the function fx mx b (with domain all real numbers) is f x m. In other words, d mx b m. dx Example Let f be the function fx 5x 4 (with domain all real numbers). Find the derivative of this function. 1 Derivatives of Monic Monomial Functions If n is a fixed positive integer, then the function fx x n (with domain all real numbers) is called a monic monomial function. (It is called a “monomial” because it has only one term and it is called “monic” because the coefficient of this term is 1.) The derivative of this function f x n x n1 . In other words, d x n n x n1 . dx This fact is sometimes referred to as the “Power Rule” of differentiation. Example Find the derivative of the monic monomial function y x 6 (with domain all real numbers). 2 The Constant Multiple and Sum Rules of Differentiation The Constant Multiple Rule of Differentiation If f is a differentiable function and c is a constant, then cf cf . Example Find the derivative of the function y 4x 6 (with domain all real numbers). 3 The Sum Rule of Differentiation If f and g are differentiable functions, then f g f g . Example Find the derivative of the function fx x 2 15. 4 Derivatives of Polynomial Functions By combining the Power, Constant Multiple, and Sum Rules of Differentiation, we can find the derivative of any polynomial function. Example For the derivative of the polynomial function fx 3x 2 6x 24 (with domain all real numbers). A graph of f is shown below. Sketch the graph of f below the graph of f and convince yourself that these graphs make sense in relation to each other. -5 -4 -3 -2 -1 1 2 3 4 5 Graph of fx 3x 2 6x 24 -5 -4 -3 -2 -1 1 2 3 4 5 Graph of f x ________________ 5 Example Find the derivative of the polynomial function y 6x 10 4x 7 2x 6 x 3 6x 865. The Generalized Power Rule of Differentiation Although we cannot prove it right now, the Power Rule d x n n x n1 dx is true even if the number n is not a positive integer. In fact, n can be any real number. Example Find the derivatives of the functions fx x , fx 1/x, and fx 4/x 3 . (Hint: Each of these functions is really a “power” function in disguise.) 6 Derivatives of Exponential Functions If a is positive constant, then the function with domain all real numbers defined by fx a x is called the base a exponential function. For example, the base 2 exponential function is fx 2 x . The most important and useful exponential function (at least for Calculus purposes) is the base e exponential function, fx e x where e is a certain irrational number that is approximately equal to 2. 71828. We will see in a moment why base e is the preferable choice of a base when using exponential functions to solve Calculus problems. (The reason, it turns out, is that its derivative is “nicer” than the derivative of any other exponential function.) The first thing we will do will be to observe that the derivative of an exponential function, fx a x , at any real number x is actually a constant multiple of a x . Furthermore, the constant that multiplies a x is f 0. In other words, if f is the function fx a x and x is any real number, then f x f 0 a x . Here is the reason: For the function fx a x and for a fixed real number x, we have fx h fx f x lim h0 h xh x lim a a h0 h x h x lim a a a h0 h h lim a 1 a x . h0 h x Since a is a constant, we may “pull it outside” of the above limit to obtain h f x lim a 1 a x . h0 h Finally, we observe that h 0h 0 f0 h f0 f 0. lim a 1 lim a a lim h0 h0 h0 h h h Thus f x f 0 a x . 7 Example For the exponential function fx 2 x , use numerical approximation (on your calculator) to estimate the value of f 0. Then write an approximate formula for the derivative of the function fx 2 x . Answer: After doing some computations, we estimate that f 0 __________. An approximate formula for the derivative of f is d 2 x _____________. dx 8 Example For the exponential function fx 3 x , use numerical approximation (on your calculator) to estimate the value of f 0. Then write an approximate formula for the derivative of the function fx 3 x . Answer: After doing some computations, we estimate that f 0 __________. An approximate formula for the derivative of f is d 3 x _____________. dx 9 For any exponential function, fx a x , we have f x f 0 a x . The number f 0 depends on the base (a) that is being used. Furthermore, the number f 0 increases (in continuous fashion) as the base a increases. We have seen that if a 2, then f 0 0. 69, and that if a 3, then f 0 1. 098 6. This means that there is a base, which we call e, that lies between 2 and 3 such that f 0 is exactly equal to 1. The number e is defined in this way. In other words, the number e is the number – the only number – such that h lim e 1 1. h0 h We can now see what makes base e so “nice” for Calculus purposes: d e x 1 e x e x . dx In other words, the derivative of the function fx e x is simply the function itself! Can you think of any other functions that are equal to their derivatives? 10