# Math Help - derivative of rational/trigonometric functions

1. ## derivative of rational/trigonometric functions

I was hoping someone could check my solution.

1. Calculate the derivative of: $\frac{\cos x}{\ x-1}$

$f(x)=(cosx)(1-x)^{-1}$

$f'(x)=(cosx)'(1-x)^{-1} + (cosx)((1-x)^{-1})'$

$f'(x)=-sinx(1-x)^{-1} + cosx(-1)(1-x)^{-2}(-1)$

$f'(x)=\frac{\-sinx}{\ 1-x} + \frac {\ cosx}{x^2-2x+1}$

Sincerely,

Raymond MacNeil

2. ## Re: derivative of rational/trigonometric functions

The step in the middle had a typo or two, but it's ok.

3. ## Re: derivative of rational/trigonometric functions

Oh, you mean I forgot to make 1 a negative exponent right? I see.

4. ## Re: derivative of rational/trigonometric functions

Originally Posted by raymac62
I was hoping someone could check my solution.

1. Calculate the derivative of: $\frac{\cos x}{\ x-1}$

$f(x)=(cosx)(1-x)^{-1}$

$f'(x)=(cosx)'(1-x)^{-1} + (cosx)((1-x)^{-1})'$

$f'(x)=-sinx(1-x)^{-1} + cosx(-1)(1-x)^{-2}(-1)$

$f'(x)=\frac{\-sinx}{\ 1-x} + \frac {\ cosx}{x^2-2x+1}$

\displaystyle \begin{align*} \frac{d}{dx}\left(\frac{\cos{x}}{x - 1}\right) &= \frac{(x - 1)\frac{d}{dx}(\cos{x}) - \cos{x}\frac{d}{dx}(x - 1)}{(x - 1)^2} \\ &= \frac{-(x - 1)\sin{x} - \cos{x}}{(x - 1)^2} \end{align*}