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# Ch4_7 - 1 Section 4.7 Numerical Differentiation...

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Next Previous Section 4.7: Numerical Differentiation 11 Section 4.7: Numerical Differentiation ® Differentiation ® Analytical ® Differentiation of a function yields the derivative ® Derivative – the slope of the function ® Numerical ® Motivation – why? ® Forward, central and backward difference ® An example

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Next Previous 22 ® Determining the slope of a function f(x) at a point x ® As an example, consider the function f(x)= x2+2x ® Assess the slope over a small change & x x x f x x f x x f 0 0 - + = ) ( ) ( ) ( Analytical derivative – the slope at a point x f(x ) f(x f(xo+ x) xo+ x slop e ( 29 ( 29 [ ] [ ] x x 2 x x x 2 x x x x f x x f x x f dx x df 0 2 0 0 2 0 0 0 0 x 0 x + - + + + = - + = = ) ( ) ( lim ) ( lim ) ( [ ] [ ] x x 2 x x x 2 x x 2 x x 2 x 2 x x x 2 x dx x df 2 0 0 x 0 2 0 0 2 0 2 0 0 x + + = - - + + + + = lim lim ) ( [ ] 2 x 2 2 x 2 x x 2 x x x 2 dx x df 0 0 2 0 0 x + = + + = + + = lim ) ( ( x 0
Next Previous Section 4.7: Numerical Differentiation 33 0 bmx anx dx ) x ( df 1 m 1 n + + = - - c bx ax ) x ( f m n + + = ) sin( ) ( ax x f = ) cos( ) ( ax a dx x df = ) cos( ) ( ax x f = ) sin( ) ( ax a dx x df - = ax e x f = ) ( ax ae dx x df = ) ( ) ln( ) ( ax x f = x a dx x df = ) ( Analytical derivative rules for simple functions

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Next Previous Section 4.7: Numerical Differentiation 44 -6 -4 -2 0 2 4 6 8 10 12 3 4 5 6 7 2 x x 54 . 0
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Ch4_7 - 1 Section 4.7 Numerical Differentiation...

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