derivative of rational/trigonometric functions

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August 25th 2011, 05:59 PM
raymac62

derivative of rational/trigonometric functions

I was hoping someone could check my solution.

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

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

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

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

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

Thanks in advance.

Sincerely,

Raymond MacNeil
August 25th 2011, 06:16 PM
TheChaz

Re: derivative of rational/trigonometric functions

The step in the middle had a typo or two, but it's ok.
August 25th 2011, 06:19 PM
raymac62

Re: derivative of rational/trigonometric functions

Oh, you mean I forgot to make 1 a negative exponent right? I see.
August 25th 2011, 07:11 PM
Prove It

Re: derivative of rational/trigonometric functions

Quote:

Originally Posted by raymac62

I was hoping someone could check my solution.

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

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

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

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

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

Thanks in advance.

Sincerely,

Raymond MacNeil

The Quotient Rule is quite straightforward...

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

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