anonymous
  • anonymous
derive the funtion.
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
anonymous
  • anonymous
\[f(x)=\frac{\sin^2(x^2+5)}{\sqrt{x^2+1}}\]? like that? will be an algebra nightmare your math teacher must hate you
anonymous
  • anonymous
haha, that's right. I was supposed t do it in class, but i got to the middle and got fed up. Can you help me? please..

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anonymous
  • anonymous
\[(\frac{f}{g})'=\frac{gf'-f'g}{g^2}\] with \[f(x)=\sin^2(x^2+5),g(x)=\sqrt{x^2+1}\] we need both \(f'\) and \(g'\) and both require the chain rule
anonymous
  • anonymous
so first we note that \(f\) is actually a composition of three functions, \(x^2+5\) , sine and the squaring function so good \[f'(x)=2\sin(x^2+5)\times \cos(x^2+5)\times 2x=4x\sin(x^2+5)\cos(x^2+5)\]
anonymous
  • anonymous
and also by the chain rule \[g'(x)=\frac{2x}{2\sqrt{x^2+1}}=\frac{x}{\sqrt{x^2+1}}\] so far so good?
anonymous
  • anonymous
yeees. :)
anonymous
  • anonymous
now we have to put this mess together for \[(\frac{f}{g})'=\frac{gf'-f'g}{g^2}\] the denominator is easy enough, it is just \(x^2+1\) but the numerator is going to be a colossal mess i guess we can just write it down
anonymous
  • anonymous
numerator is \[\sqrt{x^2+1}\times 4x\sin(x^2+5)\cos(x^2+5)-\sin^2(x^2+5)\times \frac{x}{\sqrt{x^2+1}}\]
anonymous
  • anonymous
and so your whole mess is \[\frac{\sqrt{x^2+1}\times 4x\sin(x^2+5)\cos(x^2+5)-\sin^2(x^2+5)\times \frac{x}{\sqrt{x^2+1}}}{x^2+1}\]
anonymous
  • anonymous
now you can get rid of the compound fraction by multiplying top and bottom by \(\sqrt{x^2+1}\) to get \[\frac{(x^2+1)\times 4x\sin(x^2+5)\cos(x^2+5)-\sin^2(x^2+5)\times x}{(x^2+1)\sqrt{x^2+1}}\] and maybe you can even clean this mess up with some more algebra by multiplying out in the numerator or something, but my algebra is not that good
anonymous
  • anonymous
omg, if i could give you a million medals i would.
anonymous
  • anonymous
there is also an annoying trig identity you can use since \[2\sin(x)\cos(x)=\sin(2x)\] so if you want to be really clever you can replace \[4x\sin(x^2+5)\cos(x^2+5)\] by \[2x\sin(2x^2+10)\] but i wouldn't
anonymous
  • anonymous
one is more than enough yw
anonymous
  • anonymous
oh also you can write the denominator as \((x^2+1)^{\frac{3}{2}}\) but that strikes me as unnecessary also
anonymous
  • anonymous
but he wants it like that lol thank you :)

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