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The Derivative
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The expression for the slope of the tangent line lim f (x + h) f (x) h
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determines a number which depends on x. Thus, the expression de nes a function, called the derivative of f . De nition: The derivative f of f is the function de ned by the formula f (x) = lim f (x + h) f (x) h
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notation There are other notations traditionally used for the derivative: Dx f (x) and dy dx dy . We shall use whichever notation is most convenient dx
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When a variable y represents f (x), the derivative is denoted by y , Dx y, or or customary in a given case.
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The derivative is so important in all parts of pure and applied mathematics that we must devote a great deal of effort to nding formulas for the derivatives of various kinds of functions. If the limit in the above de nition exists, the function f is said to be differentiable at x, and the process of calculating f is called differentiation of f . EXAMPLES
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(a) Let f (x) = 3x + 5 for all x. Then, f (x + h) = 3(x + h) + 5 = 3x + 3h + 5 f (x + h) f (x) = (3x + 3h + 5) (3x + 5) = 3x + 3h + 5 3x 5 = 3h 3h f (x + h) f (x) = =3 h h Hence, f (x) = lim
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f (x + h) f (x) = lim 3 = 3 h h 0
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or, in another notation, Dx (3x + 5) = 3. In this case, the derivative is independent of x. 97
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THE DERIVATIVE
[CHAP. 12
(b) Let us generalize to the case of the function f (x) = Ax + B, where A and B are constants. Then, f (x + h) f (x) [A(x + h) + B] (Ax + B) Ax + Ah + B Ax B Ah = = = =A h h h h f (x + h) f (x) = lim A = A f (x) = lim h h 0 h 0
Thus, we have proved:
Theorem 12.1: Dx (Ax + B) = A By letting A = 0 in Theorem 12.1, we obtain: Corollary 12.2: Dx (B) = 0; that is, the derivative of a constant function is 0. Letting A = 1 and B = 0 in Theorem 12.1, we obtain: Corollary 12.3: Dx (x) = 1 By the computations in Problems 11.1, 11.2, and 11.3(a), we have: Theorem 12.4: (i) Dx (x 2 ) = 2x (ii) Dx (x 3 ) = 3x 2 (iii) Dx 1 x = 1 x2
We shall need to know how to differentiate functions built up by arithmetic operations on simpler functions. For this purpose, several rules of differentiation will be proved. RULE 1. (i) Dx ( f (x) + g(x)) = Dx f (x) + Dx g(x) The derivative of a sum is the sum of the derivatives. (ii) Dx ( f (x) g(x)) = Dx f (x) Dx g(x) The derivative of a difference is the difference of the derivatives. For proofs of (i) and (ii), see Problem 12.1(a). EXAMPLES
(a) Dx (x 3 + x 2 ) = Dx (x 3 ) + Dx (x 2 ) = 3x 2 + 2x (b) Dx x 2 1 x = Dx (x 2 ) Dx 1 x 1 = 2x 2 x 1 = 2x + 2 x
RULE 2. Dx (c f (x)) = c Dx f (x) where c is a constant. For a proof, see Problem 12.1(b).
EXAMPLES
(a) Dx (7x 2 ) = 7 Dx (x 2 ) = 7 2x = 14x (b) Dx (12x 3 ) = 12 Dx (x 3 ) = 12(3x 2 ) = 36x 2 (c) Dx 4 x = Dx ( 4) 1 x = 4 Dx 1 x 1 = 4 2 x 4 = 2 x
(d) Dx (3x 3 + 5x 2 + 2x + 4) = Dx (3x 3 ) + Dx (5x 2 ) + Dx (2x) + Dx (4) = 3 Dx (x 3 ) + 5 Dx (x 2 ) + 2 Dx (x) + 0 = 3(3x 2 ) + 5(2x) + 2(1) = 9x 2 + 10x + 2
CHAP. 12]