Chapter3 - Section 3.1 Derivatives of Polynomials and...

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1 Section 3.1 Derivatives of Polynomials and Exponential Functions • Goals – Learn formulas for the derivatives of • Constant functions • Power functions • Exponential functions – Learn to find new derivatives from old: • Constant multiples •Sums and differences Constant Functions • The graph of the constant function f ( x ) = c is the horizontal line y = c – which has slope 0 , – so we must have f ( x ) = 0 (see the next slide). • A formal proof is easy: Power Functions • Next we look at the functions f ( x ) = x n , where n is a positive integer. • If n = 1 , then the graph of f ( x ) = x is the line y = x , which has slope 1 , so f ( x ) = 1. • We have already seen the cases n = 2 and n = 3 : Power Functions (cont’d) •Fo r n = 4 we find the derivative of f ( x ) = x 4 as follows: Power Functions (cont’d) • There seems to be a pattern emerging! • It appears that in general, if f ( x ) = x n , then f ( x ) = nx n -1 . • This turns out to be the case: Power Functions (cont’d) • We illustrate the Power Rule using a variety of notations: • It turns out that the Power Rule is valid for any real number n , not just positive integers:
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2 Power Functions (cont’d) Constant Multiples • The following formula says that the derivative of a constant times a function is the constant times the derivative of the function : Sums and Differences • These next rules say that the derivative of a sum (difference) of functions is the sum (difference) of the derivatives : Example Exponential (cont’d) • Geometrically, this means that – of all the exponential functions y = a x , – the function f ( x ) = e x is the one whose tangent at (0, 1) has a slope f (0) that is exactly 1 . – This is shown on the next slide: Exponential (cont’d)
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3 Exponential (cont’d) • This leads to the following differentiation formula: • Thus, the exponential function f ( x ) = e x is its own derivative . Example • If f ( x ) = e x x , find f ( x ) and f ′′ (0) . •So lu t ion The Difference Rule gives • Therefore Solution (cont’d) • Note that e x is positive for all x , so f ′′ ( x ) > 0 for all x . • Thus, the graph of f is concave up. – This is confirmed by the graph shown. Review • Derivative formulas for polynomial and exponential functions • Sum and Difference Rules • The natural exponential function e x
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1 Section 3.2 The Product and Quotient Rules • Goals – Learn formulas for the derivatives of the •p roduc t and • quotient of two functions whose derivatives are known The Product Rule • Suppose that f ( x ) and g ( x ) are each known; what is ( fg ) ( x ) ? • It is tempting to suppose that ( fg ) ( x ) = f ( x ) g ( x ) • But this is wrong ! – Try it with f ( x ) = x and g ( x ) = x 2 , for example. Product Rule (cont’d) • To see the correct formula, we first assume that u = f ( x ) and v = g ( x ) are both positive differentiable functions.
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Chapter3 - Section 3.1 Derivatives of Polynomials and...

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