I have tried 2 different approaches, both yielding different results and both results are present in the options. Am I doing something wrong or why is this happening?

stropa0u
2022-01-24
Answered

If $y=\mathrm{arctan}\left(\frac{2x-1}{1+x-{x}^{2}}\right)$ , then $\frac{dy}{dx}$ at x=1 is equal to?

I have tried 2 different approaches, both yielding different results and both results are present in the options. Am I doing something wrong or why is this happening?

I have tried 2 different approaches, both yielding different results and both results are present in the options. Am I doing something wrong or why is this happening?

You can still ask an expert for help

bekiffen32

Answered 2022-01-25
Author has **8** answers

In the derivative of

$y=\mathrm{arctan}\left(x\right)-\mathrm{arctan}(1-x)+k\pi$

$\frac{dy}{dx}=\frac{1}{1+{x}^{2}}-(-\frac{1}{1+{(1-x)}^{2}})$

which is the same as in the second approach.

Also note that the inverse tangent is an odd function, so that$\mathrm{arctan}(1-x)=-\mathrm{arctan}(x-1)$ , so that there is no structural difference between the two approaches

which is the same as in the second approach.

Also note that the inverse tangent is an odd function, so that

Fallbasisz8

Answered 2022-01-26
Author has **9** answers

A computationally simpler method of directly calculating the derivative at x=1 is to write

$\mathrm{tan}y=\frac{2x-1}{1+x-{x}^{2}}$

so that

$\mathrm{log}\mathrm{tan}y=\mathrm{log}(2x-1)-\mathrm{log}(1+x-{x}^{2})$

Then implicit differentiation yields

$\frac{{\mathrm{sec}}^{2}y}{\mathrm{tan}y}\frac{dy}{dx}=\frac{2}{2x-1}-\frac{1-2x}{1+x-{x}^{2}}$

Since, when x=1, we have$\mathrm{tan}y=\frac{2-1}{1+1-1}=1$ , it follows from the trigonometric identity ${\mathrm{sec}}^{2}y=1+{\mathrm{tan}}^{2}y\text{}\text{that}\text{}{\mathrm{sec}}^{2}y=1+{1}^{2}=2$ ; therefore

$2{\left[\frac{dy}{dx}\right]}_{x=1}=\frac{2}{2-1}-\frac{1-2}{1+1-1}=3$

hence the answer is 3/2. Note this solution uses two tactics; logarithmic differentiation and implicit differentiation, to make the calculation extremely simple.

so that

Then implicit differentiation yields

Since, when x=1, we have

hence the answer is 3/2. Note this solution uses two tactics; logarithmic differentiation and implicit differentiation, to make the calculation extremely simple.

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