lec3 - 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 3 18.01 Fall 2006 Lecture 3 (presented by Kobi Kremnizer): Derivatives of Products, Quotients, Sine, and Cosine Derivative Formulas There are two kinds of derivative formulas: d 1. Specific Examples: -x" or - dx 2. General Examples: (u + v)' = u1 + v1 and (cu) = cul (where c is a constant) A notational convention we will use today is: Proof of (u + v) = u' + v'. (General) $@rt by using the definition af &hederivative. (U + v)'(x) = lirn (U + U)(X + ax) - + v)(x) Ax-0 ax U(X + + V(X + AX) - U(X) - V(X) = = lim { u(x + Ax) - u(x) + v (x + - v(x) AX-O Ax Ax Follow the same procedure to prove that (cu)' = cu'. Derivatives of sin x and cos x. (Specific) Last time, we computed sinx lim - = 1 x-0 x d sin(0 + - sin(0) sin( Ax) -(sinx) IZEo = = lim - = 1 dx AX-o AX-o d COS(O + - COS(O) COS(~X) i - -(co~x)I~=~ = = lim = 0 dx AX-0 AZ-O d d So, we know the value of - sin x and of - cos x at x = 0. Let us find these for arbitrary x.
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lec3 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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