Intro to Analysis in-class_Part_35

# Intro to Analysis in-class_Part_35 - Let y = f x and s = f...

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5.2 Properties of the derivative 75 which is always valid for some x ( s,t ).

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76 Math 413 Diﬀerentiation 5.3 Calculus of derivatives 5.3.1 Arithmetic rules Theorem 5.3.1 (Linearity) . f,g diﬀerentiable at x = ( af + bg ) 0 ( x ) = af 0 ( x ) + bg 0 ( x ) , a,b R . Proof. HW (follows immediately from limit defn) Theorem 5.3.2 (Product rule) . f,g diﬀerentiable at x = ( fg ) 0 ( x ) = f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) . Proof. Let h = fg so that h ( t ) - h ( x ) = f ( t ) g ( t ) - f ( t ) g ( x ) + f ( t ) g ( x ) - f ( x ) g ( x ) h ( t ) - h ( x ) t - x = f ( t )[ g ( t ) - g ( x )] t - x + g ( x )[ f ( t ) - f ( x )] t - x . Theorem 5.3.3 (Quotient rule) . f,g diﬀerentiable at x , g ( x ) 6 = 0 = f g · 0 ( x ) = f 0 ( x ) g ( x ) - f ( x ) g 0 ( x ) g 2 ( x ) . Proof. HW: Let h = f/g so that h ( t ) - h ( x ) t - x = 1 g ( t ) g ( x ) g ( x ) f ( t ) - f ( x ) t - x - f ( x ) g ( t ) - g ( x ) t - x . Theorem 5.3.4 (Chain Rule) . If f is diﬀerentiable at x and g is diﬀerentiable at f ( x ) , then g f is diﬀerentiable at x with ( g f ) 0 ( x ) = g 0 ( f ( x )) f 0 ( x ) . Proof.
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Unformatted text preview: Let y = f ( x ) and s = f ( t ) and deﬁne h ( t ) = g ( f ( t )). By the defn of derivative, f ( t )-f ( x ) = ( t-x )[ f ( x ) + o (1)] as t → x g ( s )-g ( y ) = ( s-y )[ g ( y ) + o (1)] as s → y. So : h ( t )-h ( x ) = g ( f ( t ))-g ( f ( x )) = ( s-y )[ g ( y ) + o (1)] = [ f ( t )-f ( x )][ g ( f ( x )) + o (1)] = [( t-x )[ f ( x ) + o (1)]][ g ( f ( x )) + o (1)] h ( t )-h ( x ) t-x = [ f ( x ) + o (1)][ g ( f ( x )) + o (1)] ....
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Intro to Analysis in-class_Part_35 - Let y = f x and s = f...

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