document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=486705>Question 486705</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the definition of the derivation to show that (d/dx)(cos x)= -sin x.<BR>\n" );
document.write( "You can use the following 3 facts:<BR>\n" );
document.write( "lim h->0 (sin h)/h =1\r<BR>\n" );
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document.write( "lim h->0 (cos h -1) / h = 0\r<BR>\n" );
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document.write( "cos(&#952; + )= cos&#952;cos - sin&#952;sin </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #332758 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=332758\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Use the definition of derivative using limits:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{d}{dx} (\cos(x)) = \lim_{\Delta x \to 0} \frac{\cos(x+\Delta x) - \cos(x)}{\Delta x}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE = \lim_{\Delta x \to 0} \frac{\cos x \cos(\Delta x) - \sin x \sin (\Delta x) - \cos x}{\Delta x} = \lim_{\Delta x \to 0} \frac{\cos x (\cos (\Delta x) - 1)}{\Delta x} - \frac{\sin x \sin(\Delta x)}{\Delta x}\">\r<BR>\n" );
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document.write( "Applying our known facts, we can treat cos x and sin x as constants and say that the first limit collapses to zero, and the second limit collapses to -sin x. Hence,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{d}{dx} (\cos x) = -\sin x\">\r<BR>\n" );
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document.write( "We could also use L'Hopital's rule to evaluate the limits, but that assumes we already know the derivative of cos x, so we shouldn't use it right now. </SPAN>\n" );
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