Question

Can this be simplified any further?

I was asked to find the derivative of f(x)=cos (x^2 -1)^1/2

I got the result -sin(1/2)(x^2 -1)^-1/2(2x)

Answer 1

This might be clearer

Original equation: \[\cos \sqrt{x^2-1}\]

my result so far: \[-\sin*1/2(x^2-1)^{-1/2}(2x)\]

Answer 2

\[\cos^{1/2}(x^2-1)\] so this

Answer 3

WAIT

Answer 4

Oh nvm ok I see... alright same as before you will need chain rule

Answer 5

I'm sorry the equation I provided in the original question is incorrect. I commented the correct problem though.

Answer 6

Ok I got ya, so we have \[f(x) = \cos(\sqrt{x^2-1)}\] I like to work with exponents as it's clearer so lets just say it's \[f(x) = \cos(x^2-1)^{1/2}\] then we have \[f'(x) = - \sin(x^2-1)^{1/2} \times ((x^2-1)^{1/2})'\] and taking that derivative we get \[f'(x) = -\frac{ \sin(x^2-1)^{1/2} }{ (x^2-1)^{1/2} }\] the 1/2 and 2 get cancelled

Answer 7

\[f'(x)=-\sin(x^2-1)^{1/2} (1/2)(x^2-1)^{-1/2}(2x)\]

I got this when I did it a second time and I think that's a little different.

Answer 8

Even if the 1/2 and the 2 got cancelled I think I still have an x left over...

Answer 9

You found a mistake, I forgot to put and x in the numerator \[f'(x) = -\frac{ \color{red}x \sin(x^2-1)^{1/2} }{ (x^2-1)^{1/2} } \]

Answer 10

oh cool I did something right!

Answer 11

Thank you so much!

Answer 12

Anytime :)