What is the derivative of $f\left(x\right)=\mathrm{cot}\left(x\right)$ ?

Lorraine Harvey
2021-12-18
Answered

What is the derivative of $f\left(x\right)=\mathrm{cot}\left(x\right)$ ?

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macalpinee3

Answered 2021-12-19
Author has **29** answers

The quotion rule:

So, we have,

Thus,

Natalie Yamamoto

Answered 2021-12-20
Author has **22** answers

Derivative of $\mathrm{cot}\left(x\right)$ is equal to $-{\mathrm{csc}}^{2}\left(x\right)$ .

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Minimizing $4{\mathrm{sec}}^{2}\left(x\right)+9{\mathrm{csc}}^{2}\left(x\right)$ for x in the first quadrant. Discrepancy in solution

Using derivatives, I am able to show that the minimum value of the expression-in-question is equal to 25. I also verified this with Desmos graphing app. However, when I tried doing this using basic algebra, the answer turns out to be 26. I do not know why this is happening, but I should be extremely grateful to you if you can point out the error.

$4{\mathrm{sec}}^{2}x+9{\mathrm{csc}}^{2}x=$

${(2\mathrm{sec}x-3\mathrm{csc}x)}^{2}+12\mathrm{sec}x\mathrm{csc}x$

The value of the above expression will be minimum when that expression inside parenthesis equals zero; and that happens when$\mathrm{tan}x=\frac{32}{}$ . Using this we can say that $\mathrm{sec}x=\frac{\sqrt{13}}{2}$ and $\mathrm{csc}x=\frac{\sqrt{13}}{3}$

If now we substitute these values in the above expression, answer turns out to be 26. I don't know why this is happening, but please help me find the error in this approach.

Using derivatives, I am able to show that the minimum value of the expression-in-question is equal to 25. I also verified this with Desmos graphing app. However, when I tried doing this using basic algebra, the answer turns out to be 26. I do not know why this is happening, but I should be extremely grateful to you if you can point out the error.

The value of the above expression will be minimum when that expression inside parenthesis equals zero; and that happens when

If now we substitute these values in the above expression, answer turns out to be 26. I don't know why this is happening, but please help me find the error in this approach.

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