Quotient Trig Example

[Jason76]

Post Edited

\(\displaystyle f(x) = \dfrac{8x}{1 - \cot(x)}\)

Using Quotient Rule:

Given: \(\displaystyle \dfrac{f(x)}{g(x)}\)

\(\displaystyle f(x) = \dfrac{g(x)[f'(x)] - f(x)[g'(x)]}{g^{2}}\)

\(\displaystyle f'(x) = \dfrac{[1 - \cot(x)](\dfrac{d}{dx} 8x) - (8x)[\dfrac{d}{dx}1 - \cot(x)]}{(1 - \cot(x))^{2}}\)

\(\displaystyle f'(x) = \dfrac{[1 - \cot(x)](8) - 8x \csc^{2}(x)}{(1 - \cot(x))^{2}}\)

\(\displaystyle f'(x) = \dfrac{[1 - \cot(x)](8) - 8x \csc^{2}(x)}{(1 - \cot(x))^{2}}\)

\(\displaystyle f'(x) = \dfrac{8 - 8 \cot(x) - 8x \csc^{2}(x)}{(1 - \cot(x))^{2}}\) - Answer

 

[Deleted member 4993]

\(\displaystyle f(x) = \dfrac{8x}{1 - \cot(x)}\)

\(\displaystyle \displaystyle \left [f'(x) = \dfrac{\dfrac{d}{dx}8x}{\dfrac{d}{dx} 1 - \cot(x)}]\right ]\).... This is TOTALLY wrong - erase it.

Using Quotient Rule:

Given: \(\displaystyle \dfrac{f(x)}{g(x)}\)

\(\displaystyle \dfrac{g(x)[f'(x)] - f(x)[g'(x)]}{g^{2}}\)

\(\displaystyle \dfrac{[1 - \cot(x)](8) - 8x + \csc^{2}(x)}{(1 - \cot(x))^{2}}\)........Incorrect - should be \(\displaystyle \dfrac{[1 - \cot(x)](8) - 8x * \csc^{2}(x)}{(1 - \cot(x))^{2}}\)

\(\displaystyle \dfrac{8 - 8 \cot(x) - 8x + \csc^{2}(x)}{(1 - \cot(x))^{2}}\) ...............Incorrect - as explained above

- Answer

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[Jason76]

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Original Post Edited.

 

[lookagain]

Post Edited

\(\displaystyle f(x) = \dfrac{8x}{1 - \cot(x)}\)

Using Quotient Rule:

Given: \(\displaystyle \dfrac{f(x)}{g(x)}\)

\(\displaystyle f(x) = \dfrac{g(x)[f'(x)] - f(x)[g'(x)]}{g^{2}} \ \ \ \) <----- Stop writing the denominator this way. \(\displaystyle \ \ \)\(\displaystyle \ \ \) It's wrong, given the context of the forms of the given.\(\displaystyle \ \ \) [g(x)]^2 is a way. **

** \(\displaystyle \ \ [g(x)]^2 \ \ is \ \ another \ \ way.\)

Edit: And another error is you can't use "f(x)" to refer to the original function and also the numerator of the function at the same time.