| Lingyis said: i'd rather not respond, but since a wrong answer has been given, i should probably correct that person. but first, i imagine "modulus" is just a synonym for "absolute value", since "modulus" to me means the magnitude of a complex number, in which case the symbol "z" is more often used than "x". i'll proceed assuming modulus means absolute value. to approach this problem, first plot the function. this is fairly easy, actually: you know how to plot f(x)=x, and you know how to plot f(x)=sin(x). visually multiply the two, and you get f(x)=x sin(x). going from sin(x) to sin(8 x) is easy, you just "squish" your graph 8 times in the horizontal direction. to get the |sin(8 x)| part right, just remember that the |...| means you always end up with a positive value. so for x |sin(8 x)|, which is the function x multiplied by sin(8 x), when x>0, this function is positive. when x<0, this function is negative. now you should know what the graph looks like. it's basically a increasing sinusoidal with bumps. it's easy to tell what points are not differentiable once you have the graph in front of you. all the "flipping" points, i.e. n*Pi/8 for all n except n=0, are not differentiable. the point n=0 is differentiable, with a derivative of 0. you're probably not required to prove it, but you'll get extra credit if you prove it. basically, prove that the left derivative and the same as the right derivative. if you don't know what i'm saying you'll certainly not required to prove it. now onto the actual derivative. the |...| is a distraction, you should first get your "domains" right (you should know that domain is a fancy word for where your function is defined on). here goes: |sin(8x)| is: sin(8x) when n*Pi/8 < x < (n+1)*Pi/8 for n = even -sin(8x) when n*Pi/8 < x < (n+1)*Pi/8 for n= odd
so taking derivative of x sin(8x) or -x sin(8x) on their appropriate domains is simply sin(8x) + 8x cos(8x) when n*Pi/8 < x < (n+1)*Pi/8 for n = even -sin(8x) - 8x cos(8x) when n*Pi/8 < x < (n+1)*Pi/8 for n= odd and is = 0 when x=0, and undefined for all x = n*Pi/8 except for n=0.
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Wasn't this my answer? at least I got the derivative right!
