Derivatives of Constant and Linear Functions

Transcription

1 These notes closely follow the presentation of the material given in James Stewart s textbook Calculus, Concepts an Contexts (2n eition). These notes are intene primarily for in-class presentation an shoul not be regare as a substitute for thoroughly reaing the textbook itself an working through the exercises therein. Derivatives of Constant an Linear Functions If c is a constant, then the erivative of the function f x c (with omain all real numbers) is f x 0. In other wors, c 0. x Example Let f be the function f x 13 (with omain all real numbers). Fin the erivative of this function. If m an b are constants, then the erivative of the function f x mx b (with omain all real numbers) is f x m. In other wors, mx b m. x Example Let f be the function f x 5x 4 (with omain all real numbers). Fin the erivative of this function. 1

2 Derivatives of Monic Monomial Functions If n is a fixe positive integer, then the function f x x n (with omain all real numbers) is calle a monic monomial function. (It is calle a monomial because it has only one term an it is calle monic because the coefficient of this term is 1.) The erivative of this function f x n x n 1. In other wors, x xn n x n 1. This fact is sometimes referre to as the Power Rule of ifferentiation. Example Fin the erivative of the monic monomial function y x 6 (with omain all real numbers). 2

3 The Constant Multiple an Sum Rules of Differentiation The Constant Multiple Rule of Differentiation If f is a ifferentiable function an c is a constant, then cf cf. Example Fin the erivative of the function y 4x 6 (with omain all real numbers). 3

4 The Sum Rule of Differentiation If f an g are ifferentiable functions, then f g f g. Example Fin the erivative of the function f x x

5 Derivatives of Polynomial Functions By combining the Power, Constant Multiple, an Sum Rules of Differentiation, we can fin the erivative of any polynomial function. Example For the erivative of the polynomial function f x 3x 2 6x 24 (with omain all real numbers). A graph of f is shown below. Sketch the graph of f below the graph of f an convince yourself that these graphs make sense in relation to each other Graph of f x 3x 2 6x Graph of f x 5

6 Example Fin the erivative of the polynomial function y 6x 10 4x 7 2x 6 x 3 6x 865. The Generalize Power Rule of Differentiation Although we cannot prove it right now, the Power Rule x xn n x n 1 is true even if the number n is not a positive integer. In fact, n can be any real number. Example Fin the erivatives of the functions f x x,f x 1/x, an f x 4/x 3. (Hint: Each of these functions is really a power function in isguise.) 6

7 Derivatives of Exponential Functions If a is positive constant, then the function with omain all real numbers efine by f x a x is calle the base a exponential function. For example, the base 2 exponential function is f x 2 x. The most important an useful exponential function (at least for Calculus purposes) is the base e exponential function, f x e x where e is a certain irrational number that is approximately equal to 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 erivative is nicer than the erivative of any other exponential function.) The first thing we will o will be to observe that the erivative of an exponential function, f x 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 wors, if f is the function f x a x an x is any real number, then f x f 0 a x. Here is the reason: For the function f x a x an for a fixe real number x, we have f x lim h 0 lim h 0 lim h 0 f x h f x h a x h a x h a x a h a x h lim a h 1 a h 0 h x. Since a x is a constant, we may pull it outsie of the above limit to obtain f x lim a h 1 a h 0 h x. Finally, we observe that lim a h 1 lim a 0 h a 0 f 0 h f 0 lim f h 0 h h 0 h h 0 h 0. Thus f x f 0 a x. 7

8 Example For the exponential function f x 2 x, use numerical approximation (on your calculator) to estimate the value of f 0. Then write an approximate formula for the erivative of the function f x 2 x. Answer: After oing some computations, we estimate that f 0. An approximate formula for the erivative of f is x 2x. 8

9 Example For the exponential function f x 3 x, use numerical approximation (on your calculator) to estimate the value of f 0. Then write an approximate formula for the erivative of the function f x 3 x. Answer: After oing some computations, we estimate that f 0. An approximate formula for the erivative of f is x 3x. 9

10 For any exponential function, f x a x, we have f x f 0 a x. The number f 0 epens on the base (a) that is being use. Furthermore, the number f 0 increases (in continuous fashion) as the base a increases. We have seen that if a 2, then f , an that if a 3, then f This means that there is a base, which we call e, that lies between 2 an 3 such that f 0 is exactly equal to 1. The number e is efine in this way. In other wors, the number e is the number the only number such that lim e h 1 1. h 0 h We can now see what makes base e so nice for Calculus purposes: x ex 1 e x e x. In other wors, the erivative of the function f x e x is simply the function itself! Can you think of any other functions that are equal to their erivatives? 10