Composite and quotient rules applied to f(x) = ln(1 + cosx)

[Guest]

The function, f, is given by
\(\displaystyle f(x) = \ln (1 + \cos x)\)

Apply the composite and quotient rules to find f'(x) and f''(x).

Please can you help?

 

[morson]

\(\displaystyle \L\ \frac{d}{dx}\[ln(g(x)] = \frac{g'(x)}{g(x)}\\), \(\displaystyle g(x) > 0\)

 

[Guest]

For f'(X), I arrived at

\(\displaystyle \frac{{ - \sin x}}{{1 + \cos x}}\)

and for f''(x) I got

\(\displaystyle - \frac{{\cos x}}{{(1 + \cos x)^2 }}\)

Is this right?

 

[galactus]

f'(x) is correct. Perhaps you could write it as \(\displaystyle \L\\-tan(\frac{x}{2})\). It's equivalent to what you have.

f''(x) is incorrect. Shoot for \(\displaystyle \L\\\frac{-1}{1+cos(x)}=\frac{-1}{2}sec^{2}(\frac{x}{2})\)