Derivative of polynomials and exponentials

# Derivative of polynomials and exponentials - h → e h-1 h...

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DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS For basic functions, we have diﬀerentiation rules as follows. 1. d dx ( c ) = 0. 2. d dx ( x ) = 1. 3. d dx ( x n ) = nx n - 1 . If we know derivatives of certain functions, we can calculate derivatives of new functions by using old ones. 4. d dx [ cf ( x )] = cf 0 ( x ). 5. d dx [ f ( x ) + g ( x )] = f 0 ( x ) + g 0 ( x ). 6. d dx [ f ( x ) - g ( x )] = f 0 ( x ) - g 0 ( x ). By using above rules, we can diﬀerentiate any polynomials. Example 1. Find The derivative of the function f ( x ) = 2 x 2 + 4 x + 1. Solution f 0 ( x ) = d dx (2 x 2 + 4 x + 1) = 2 d dx ( x 2 ) + 4 d dx ( x ) + d dx (1) = 2(2 x ) + 4(1) + 0 = 4 x + 4 Example 2. Find The derivative of the function f ( x ) = a x . Solution f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 a x + h - a x h = lim h 0 a x ( a h - 1) h = a x lim h 0 a h - 1 h In the example above, we do not know the limit lim h 0 a h - 1 h . However, if the limit is 1, then we can get a function whose derivative equals itself. Because this type of function is frequently used, we deﬁne a number a with this property as follows. Deﬁnition 1. e is the number such that lim

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Unformatted text preview: h → e h-1 h = 1. 1 2 DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS From the deﬁnition, we can get the following diﬀerentiation formula. Theorem 1. d dx ( e x ) = e x Example 3. If f ( x ) = e x-x 2 + 3 x , ﬁnd f and f 00 . Solution f ( x ) = d dx ( e x-x 2 + 3 x ) = d dx ( e x )-d dx ( x 2 ) + 3 d dx ( x ) = e x-2 x + 3 f ( x ) = d dx ( e x-2 x + 3) = d dx ( e x )-2 d dx ( x ) + d dx (3) = e x-2 Example 4. Find an equation of the normal line to the curve y = e x-x 2 + 3 x at the point P (0 , 1). Solution Since the slope of the tangent line at the point P is y (0) = 4, we can see that the slope of the normal line is-1 4 . Thus, the equation of the normal line is y-1 =-1 4 ( x-0) , or y =-1 4 x + 1 ....
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Derivative of polynomials and exponentials - h → e h-1 h...

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