lec4 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Lecture 4 Sept. 14, 2006 18.01 Fall 2006 Lecture 4 Chain Rule, and Higher Derivatives Chain Rule We’ve got general procedures for differentiating expressions with addition, subtraction, and multi- plication. What about composition? Example 1. y = f ( x ) = sin x,x = g ( t ) = t 2 . So, y = f ( g ( t )) = sin( t 2 ). To ±nd dy , write dt t = t 0 + Δ t t 0 = t 0 x = x 0 + Δ x x 0 = g ( t 0 ) y = y 0 + Δ y y 0 = f ( x 0 ) Δ y = Δ y Δ x Δ t Δ x · Δ t As Δ t 0, Δ x 0 too, because of continuity. So we get: dy dy dx = The Chain Rule! dt dx dt In the example, dx dt = 2 t and dy dx = cos x . So, d dt sin( t 2 ) ± = ( dy dx )( dx dt ) = = (cos x )(2 t ) (2 t ) cos( t 2 ) ± Another notation for the chain rule ² ³ d dt f ( g ( t )) = f ( g ( t )) g ( t ) or d dx f ( g ( x )) = f ( g ( x )) g ( x ) Example 1. (continued) Composition
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lec4 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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