alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Limit step finally we take the limit as t 0 to get

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Unformatted text preview: Let f(x) = anxn + an−1xn−1 + . . . + a2x2 + a1x1 + a0 be any polynomial of degree n, then : df(x) d = f’(x) = n .a nxn−1 + (n−1).a n−1xn−2 + . . . + 2a2x1 + a1 dx You have now learned to DIFFERENTIATE xn when the exponent n is an integral or whole number. What if n is a RATIONAL number ? 1 -- 3 How would you DIFFERENTIATE x /2 o r x /2 ? Mechanically, we may say: 1 -- 3 /2 d x /2 1 -- 1/ -- 3 -- 5 -- x 2 a nd dx =2 = -- x /2 dx dx 2 But we must be careful and add : for x > 0 The same rule applies. The proof is not required at this level. 70 15. Differentiation from FIRST PRINCIPLES Let us now find the FIRST DERIVATIVE of some simple functions by directly applying the definition. This is known as differentiation from ffirst principles. Before we differentia irst dif entiation from ir DIFFERENTIATE a function we must ensure that it is WELL-BEHAVED : SINGLE VALUED, CONTINUOUS and DIFFERENTIABLE. 1: Example 1 What is the instantaneous rate of change o f f(x) = C ? instantaneous f(x) f (x) C P1 P2 0,0) ( 0,0 ) x1 f(x) = C x2 VERAGE step 1. AVERAGE ste p : constant x P2 -- P1 = f(x2) -- f(x1) = C -- C = 0 x2 -- x1 x2 -- x1 x2 -- x1 x2 -- x1 2. TENDS TO step : since x2 only TENDS TO x1 : x2 ≠ x1, we may simplify . 0 =0 x2 -- x1 3. LIMIT step : now we may take the LIMIT by letting x2 coincide with x1 . L imit 0 = 0 x →x 2 1 , f (x) = df (x) dC = =0 dx dx chang hange Since f(x) is CONSTANT its r ate of change is zero. 71 2: Example 2 What is the instantaneous rate of change o f y = bx ? instantaneous It is not necessary to find the derivative in 3 steps. We may combine all 3 steps in one expression. bx y = y P2 2b P1 b (0 , 0 ) 2 1 x y(x 2) -- y(x1) dy -L imit b x2 -- b x 1 = xL imitx P2 P1 = xL imitx x 2 -- x 1 = x 2 → x1 x 2 -- x 1 →1 → 1 x 2 -- x 1 dx 2 2 Since x 2 only TENDS TO x 1 : x 2 = x 1, we can cancel out the ( x 2 -- x 1 ) = 0. / / We simplify first and then take the LIMIT. Observe how we may factor out the coefficient b. = xL imitx → 2...
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