let f(x)=\[cos(ax+b)\]
\[\frac{\mathrm{d}{ } }{\mathrm{d} x} cos(ax+b)=-asin(ax+b)\]=\[acos(ax+b+\frac{\Pi }{2})\]
\[\frac{\mathrm{d^2}{ } }{\mathrm{d} x^2} f(x)=-a^2cos(ax+b)=a^2cos(ax+b+\frac{2\Pi }{2})\]
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\[ \frac{\mathrm{d^n}{ } }{\mathrm{d} x^n} =a^ncos(ax+b+\frac{n\Pi }{2})\]
problem
find the 2nd derivative of cos 2x at x=\[ \frac{\Pi }{2}\]
comparing with the ax+b we get a=2 b=0
here n=2 since we have to find 2nd derivative
\[ \frac{\mathrm{d^2}{ } }{\mathrm{d} x^2} (cos2x)=2^2cos(2x+\frac{2\Pi }{2} )\]
=\[-4cos2x\]
value at x=\[ \frac{\Pi }{2}\] is 4
