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Differentiate $y=x^0\tan x^0$.

That's the main question. I simply know that anything to the power $0$ is equal to $1.$ But look what my book did.

$$y=\frac{\pi x}{180}\tan {\frac{\pi x}{180}}.$$

And, wrote as a side note $$x^0=\frac{\pi x}{180}c.$$

Is there any rule of it? It is little bit weird. I was learning $x^0=1.$ But here I am seeing something else.

Reason why I thought it was power instead of degree :

enter image description here

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    $\begingroup$ Degrees. $x$ degrees. That's $x^\circ.$ $\endgroup$ Commented Jul 6, 2021 at 3:41
  • $\begingroup$ @SeanRoberson Actually, it was looking like 0 in my book. Anyway, thanks $\endgroup$ Commented Jul 6, 2021 at 3:43
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    $\begingroup$ @Istiak Maybe you are looking for algebra-precalculus. $\endgroup$ Commented Jul 6, 2021 at 4:41
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    $\begingroup$ The typesetting in that book could be better. There is a difference between $x^\circ$ and $x^0. \qquad$ $\endgroup$ Commented Jul 6, 2021 at 6:39
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    $\begingroup$ Even with the understanding that $x$ is in degrees, I still think this is confusing. Sure, $\tan x^{\circ}$ is understandable enough even if it can be argued that it's a slight abuse of notation, but if it was $y=x^{\circ}$ instead, we're supposed to say that $y'=\pi/180$ because $^{\circ}$ actually represents a function? In other words, if you asked me "differentiate $y=x\tan x$ where $x$ is measured in degrees," I would not convert the first $x$ and thus get a different answer than your book. $\endgroup$ Commented Oct 28, 2021 at 19:09

2 Answers 2

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$$x^\circ=\frac{\pi x}{180}^{\,c}$$

As pointed out, $^\circ$ in this context is the degree symbol, not the number zero.

As well, the symbol $^c$ is the (these-days-rarely-used) symbol for radian, and stands for “circular measure”.

P.S. Notice that the conversion (pre-processing) was suggested because $$\dfrac{\mathrm{d}}{\mathrm{d}x}\tan(x^{\circ})=\frac{\pi}{180}\sec^2(x^{\circ})\\\neq\sec^2(x^{\circ}).$$

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The $ ^o$ means it is in degrees, not $0$. I know it looks confusing. Note that it multiplies by $\frac{\pi}{180}$, which is the conversion factor between degrees and radians. Now see if you can solve the problem knowing this fact.

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