Question

derive the funtion.

Answer 1

\[f(x)=\frac{\sin^2(x^2+5)}{\sqrt{x^2+1}}\]? like that?
will be an algebra nightmare
your math teacher must hate you

Answer 2

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..

Answer 3

\[(\frac{f}{g})'=\frac{gf'-f'g}{g^2}\] 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

Answer 4

so first we note that \(f\) is actually a composition of three functions, \(x^2+5\) , sine and the squaring function so good
\[f'(x)=2\sin(x^2+5)\times \cos(x^2+5)\times 2x=4x\sin(x^2+5)\cos(x^2+5)\]

Answer 5

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?

Answer 6

yeees. :)

Answer 7

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

Answer 8

numerator is
\[\sqrt{x^2+1}\times 4x\sin(x^2+5)\cos(x^2+5)-\sin^2(x^2+5)\times \frac{x}{\sqrt{x^2+1}}\]

Answer 9

and so your whole mess is \[\frac{\sqrt{x^2+1}\times 4x\sin(x^2+5)\cos(x^2+5)-\sin^2(x^2+5)\times \frac{x}{\sqrt{x^2+1}}}{x^2+1}\]

Answer 10

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

Answer 11

omg, if i could give you a million medals i would.

Answer 12

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

Answer 13

one is more than enough
yw

Answer 14

oh also you can write the denominator as \((x^2+1)^{\frac{3}{2}}\) but that strikes me as unnecessary also

Answer 15

but he wants it like that lol thank you :)