2.14 DifferentialsRecall: ( ) ( )' 'dy yf x y xdx xD= = =D all are examples of ways to express derivative and that2 12 1y y yslopex x x- D= =- DY2 Y1( ) ( )( )0limxf x x f xx x xD �+D -+D - X2 X1Back to Differentials:Definition: Let ( )y f x=represent a function that is diff’able on an open interval containing X. The differential of X (denoted by dx) is any non-zero real number.The differential of Y (denoted by dy) is ( )'dy f x dx= � Y=f(x) ( )( ),x x f x x+D +D ΔY ( )'f x x�D Δx f(x) m= rise m(run) = rise run f’(x)(Δx) = rise X X+ΔX*Note that ΔY ≈f(x+Δx)-f(x)= IF Δx is “small”( ) ( ) ( )'f x x f x f x x+ - � �DDOr ( ) ( ) ( )'f x x f x f x dx+ - �D �Or ( ) ( ) ( )'f x x f x f x dxD � ++Or ( ) ( )f x x f x dy+D � +****Calculating Differentials:Function Derivative Differential2y x=2dyxdx=2dy x dx= � sin 3y x=3cos3dyxdx=3cos3dy x dx= � ( )1221y x= +( )122211 221dyx xdxdy xdxx-= + �=+21xdy dxx= �+**Example:If Δx is small:Use differentials to approximate 16.5( ) ( ) ( )'f x x f x f x dx+D � + ( )f x x= ( )121 1'22f x xx-= =( ).516.5 16 11.562 16= + � �+1 18 24 4.0625� + ��*Note: With a calculator you get 16.5
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