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**Unformatted text preview: **Differentiation The Derivative of a Real Function 5.1 Definition Let f e defined (and real-valued) on [ a, b ]. For any x ∈ [ a, b ] form the quotient φ ( x ) = f ( t ) − f ( x ) t − x ( a < t < b, t negationslash = x ) , and define f ′ ( t ) = lim t → x φ ( t ) , provided this limit exists in accordance with Definition 4.1. The function f ′ is called the derivative of f . If f ′ is defined at a point x , we say that f is differentiable at x . If f ′ is defined at every point of a set E ⊆ [ a, b ], we say that f is differentiable on E . 5.2 Theorem Let f be defined on [ a, b ] . If f is differentiable at a point x ∈ [ a, b ] then f is continuous at x . 5.3 Theorem Suppose f and g are defined on [ a, b ] and are differentiable at a point x ∈ [ a, b ] . Then f + g , fg , and f/g are differentiable at x , and (a) ( f + g ) ′ ( x ) = f ′ ( x ) + g ′ ( x ) ; (b) ( fg ) ′ ( x ) = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) ; (c) g ( x ) f ′ ( x ) − g ′ ( x ) f ( x ) g 2 ( x ) . If ( c ) , we assume of course that g ( x ) negationslash = 0 ....

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