극한·미분 심화 전체 문제

The domain of $\mathop{\mathrm{arcsec}} x$ is $|x| \geq 1$: that is, we can plug into arcsecant only values with absolute value greater than or equal to one. Since $-x^2-2 \leq -2$, every real value of $x$ gives us an acceptable value to plug into arcsecant. So, the domain of $f(x)$ is all real numbers.

To differentiate, we use the chain rule. Remember $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left\{\mathop{\mathrm{arcsec}} x\right\} = \dfrac{1}{|x|\sqrt{x^2-1}}$.

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\left\{\mathop{\mathrm{arcsec}}(-x^2-2)\right\}&= \dfrac{1}{|-x^2-2|\sqrt{(-x^2-2)^2-1}}\cdot(-2x)\\ &=\dfrac{-2x}{(x^2+2)\sqrt{x^4+4x^2+3}}. \end{aligned}$$

We use the chain rule, remembering that $a$ is a constant.

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\left\{\frac{1}{a}\arctan\left(\dfrac{x}{a}\right)\right\}&= \frac{1}{a}\cdot\frac{1}{1+\left(\frac{x}{a}\right)^2}\cdot \frac{1}{a}\\ &=\frac{1}{a^2+x^2} \end{aligned}$$

The domain of arctangent is all real numbers, so the domain of $f(x)$ is also all real numbers.

We differentiate using the product and chain rules.

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\left\{ x\arcsin x + \sqrt{1-x^2}\right\}&= \arcsin x + \frac{x}{\sqrt{1-x^2}}+ \frac{-2x}{2\sqrt{1-x^2}}\\ &=\arcsin x \end{aligned}$$

The domain of $\arcsin x$ is $[-1,1]$, and the domain of $\sqrt{1-x^2}$ is all values of $x$ so that $1-x^2 \geq 0$, so $x$ in $[-1,1]$. Therefore, the domain of $f(x)$ is $[-1,1]$.

We differentiate using the chain rule:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\{\arctan(x^2)\}&=\frac{2x}{1+x^4} \end{aligned}$$

This is zero exactly when $x=0$.

Using formulas you should memorize from this section, $$\frac{\mathrm{d}}{\mathrm{d}x}\{\arcsin x + \arccos x\}=\frac{1}{\sqrt{1-x^2}}+\frac{-1}{\sqrt{1-x^2}}=0$$

Remark: the only functions with derivative equal to zero everywhere are constant functions, so $\arcsin x + \arccos x$ should be a constant. Since $\sin \theta = \cos \left(\frac{\pi}{2}-\theta\right)$, we can set

$$\begin{aligned} \sin\theta&=x & \cos\left(\frac{\pi}{2}-\theta\right)&=x\\ \text{where $x$ and $\theta$ are the same in both expressions, and $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$. Then} \\ \arcsin x &=\theta & \arccos x &= \frac{\pi}{2}-\theta \\ \text{We note here that arcsine is the inverse of the sine function \textit{ restricted to} $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. So, since we restricted $\theta$ to this domain, $\sin \theta=x$ really does imply $\arcsin x = \theta$. (For an example of why this matters, note $\sin(2\pi)=0$, but $\arcsin (0)=0 \neq 2\pi$.) Similarly, arccosine is the inverse of the cosine function restricted to $[0,\pi]$. Since $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$, then $0 \leq (\frac{\pi}{2}-\theta) \leq \pi$, so $\cos\left(\frac{\pi}{2}- \theta\right) =x$ really does imply $\arccos x=\frac{\pi}{2}-\theta$.} \\ \end{aligned}$$

So, $$\arcsin x+\arccos x =\theta+\frac{\pi}{2}-\theta =\frac{\pi}{2}$$ which means the derivative we were calculating was actually just $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left\{\dfrac{\pi}{2}\right\}=0$.

Using the chain rule, $$y'=\frac{-{\frac{1}{ x^2}}}{\sqrt{1-\left({\frac{1}{ x}}\right)^2}}=\frac{-1}{x^2\sqrt{1-\frac{1}{x^2}}}.$$

Using the chain rule, $$y'=\frac{-{\frac{1}{ x^2}}}{1+\left({\frac{1}{ x}}\right)^2}=\frac{-1}{x^2+1}.$$

Using the product rule:

$$\begin{aligned} \displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left\{(1+x^2)\arctan x\right\} &=2x\arctan x+(1+x^2)\frac{1}{1+x^2}\\ &=2x\arctan x+1\end{aligned}$$

Let $\theta = \arctan x$. Then $\theta$ is the angle of a right triangle that gives $\tan \theta = x$. In particular, the ratio of the opposite side to the adjacent side is $x$. So, we have a triangle that looks like this:

where the length of the hypotenuse came from the Pythagorean Theorem. Now, $$\sin\left(\arctan x\right) = \sin \theta = \frac{\text{opp}}{\text{hyp}} = \frac{x}{\sqrt{x^2+1}}$$ From here, we differentiate using the quotient rule:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\left\{\frac{x}{\sqrt{x^2+1}} \right\}&= \frac{\sqrt{x^2+1}-x\frac{2x}{2\sqrt{x^2+1}}}{x^2+1}\\ &=\left(\frac{\sqrt{x^2+1}-\frac{x^2}{\sqrt{x^2+1}}}{x^2+1}\right)\cdot\frac{\sqrt{x^2+1}}{\sqrt{x^2+1}}\\ &=\frac{(x^2+1)-x^2}{(x^2+1)^{3/2}}\\ &=\frac{1}{(x^2+1)^{3/2}}=(x^2+1)^{-3/2} \end{aligned}$$

Remark: another strategy is to differentiate first, using the chain rule, then draw a triangle to simplify the resulting expression $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left\{\sin\left(\arctan x\right)\right\}=\dfrac{\cos(\arctan x)}{1+x^2}$.

Let $\theta = \arcsin x$. Then $\theta$ is the angle of a right triangle that gives $\sin \theta = x$. In particular, the ratio of the opposite side to the hypotenuse is $x$. So, we have a triangle that looks like this:

where the length of the adjacent side came from the Pythagorean Theorem. Now, $$\cot\left(\arcsin x\right) = \cot \theta = \frac{\text{adj}}{\text{opp}} = \frac{\sqrt{1-x^2}}{x}$$ From here, we differentiate using the quotient rule:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\left\{\frac{\sqrt{1-x^2}}{x} \right\}&= \frac{x\frac{-2x}{2\sqrt{1-x^2}}-\sqrt{1-x^2}}{x^2}\\ &=\frac{-x^2-(1-x^2)}{x^2\sqrt{1-x^2}}\\ &=\frac{-1}{x^2\sqrt{1-x^2}} \end{aligned}$$

Remark: another strategy is to differentiate first, using the chain rule, then draw a triangle to simplify the resulting expression $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left\{\cot\left(\arcsin x\right)\right\}=\frac{-\csc^2(\arcsin x)}{\sqrt{1-x^2}}$.

The line $y=2x+9$ has slope $2$, so we must find all values of $x$ between $-1$ and $1$ ($\arcsin x$ is only defined for these values of $x$) for which $\frac{\mathrm{d}}{\mathrm{d}x}\{\arcsin x\}=2$. Evaluating the derivative:

$$\begin{aligned} y&=\arcsin x\\ 2=y'&=\frac{1}{\sqrt{1-x^2}}\\ 4&=\frac{1}{1-x^2}\\ \frac{1}{4}&=1-x^2\\ x^2&=\frac{3}{4}\\ x&=\pm\frac{\sqrt{3}}{2}\\ (x,y)&=\pm\big(\frac{\sqrt{3}}{2},\frac{\pi}{3}\big) \end{aligned}$$

We differentiate using the chain rule:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\{\arctan(\csc x)\}&=\frac{1}{1+\csc^2x}\cdot\frac{\mathrm{d}}{\mathrm{d}x}\{\csc x\}\\ &=\frac{-\csc x \cot x}{1+\csc^2x}\\ &=\frac{-\frac{1}{\sin x}\cdot \frac{\cos x}{\sin x}}{1+\left(\frac{1}{\sin x}\right)^2}\\ &=\frac{-\cos x}{\sin^2x+1} \end{aligned}$$

So if $f'(x)=0$, then $\cos x=0$, and this happens when $x=\dfrac{(2n+1)\pi}{2}$ for any integer $n$. We should check that these points are in the domain of $f$. Arctangent is defined for all real numbers, so we only need to check the domain of cosecant; when $x=\dfrac{(2n+1)\pi}{2}$, then $\sin x=\pm1 \neq 0$, so $\csc x = \dfrac{1}{\sin x}$ exists.

Since $g(y)=f^{-1}(y)$,

$$\begin{aligned} f(g(y))&=f\left(f^{-1}(y)\right)=y \\ \text{Now, we can differentiate with respect to $y$ using the chain rule.} \\ \frac{\mathrm{d}}{\mathrm{d}y}\left\{f(g(y))\right\}&=\frac{\mathrm{d}}{\mathrm{d}y}\{y\}\\ f'(g(y))\cdot g'(y)&=1\\ g'(y)&=\frac{1}{f'(g(y))}=\frac{1}{1-\sin g(y)} \end{aligned}$$

Write $g(y)=f^{-1}(y)$. Then $g(f(x))=x$, so differentiating both sides (using the chain rule), we see

$$\begin{aligned} g'(f(x))\cdot f'(x)=1 \\ \text{What we want is $g'(\pi-1)$, so we need to figure out which value of $x$ gives $f(x)=\pi-1$. A little trial and error leads us to $x=\frac{\pi}{2}$.} \\ g'(\pi-1)\cdot f'\left(\frac{\pi}{2}\right)&=1 \\ \text{Since $f'(x)=2-\cos(x)$, $f'\left(\frac{\pi}{2}\right)=2-0=2$:} \\ g'(\pi-1)\cdot 2&=1\\ g'(\pi-1)=\frac{1}{2} \end{aligned}$$

Write $g(y)=f^{-1}(y)$. Then $g(f(x))=x$, so differentiating both sides (using the chain rule), we see

$$\begin{aligned} g'(f(x))f'(x)&=1 \\ \text{What we want is $g'(e+1)$, so we need to figure out which value of $x$ gives $f(x)=e+1$. A little trial and error leads us to $x=1$.} \\ g'(f(1))f'(1)&=1\\ g'(e+1)\cdot f'(1)&=1\\ g'(e+1) &= \frac{1}{f'(1)} \\ \text{It remains only to note that $f'(x)=e^x+1$, so $f'(1)=e+1$} \\ g'(e+1)&=\frac{1}{e+1} \end{aligned}$$

We use logarithmic differentiation, our standard method of differentiating an expression of the form $(\text{function})^{\text{function}}$.

$$\begin{aligned} f(x)&=[\sin x +2]^{\mathop{\mathrm{arcsec}} x}\\ \log(f(x))&=\mathop{\mathrm{arcsec}} x \cdot \log[\sin x +2]\\ \frac{f'(x)}{f(x)}&=\frac{1}{|x|\sqrt{x^2-1}}\log[\sin x +2]+\mathop{\mathrm{arcsec}} x \cdot \frac{\cos x}{\sin x +2}\\ f'(x)&=[\sin x +2]^{\mathop{\mathrm{arcsec}} x}\left(\frac{\log[\sin x +2]}{|x|\sqrt{x^2-1}}+ \frac{\mathop{\mathrm{arcsec}} x \cdot\cos x}{\sin x +2}\right) \end{aligned}$$

The domain of $\mathop{\mathrm{arcsec}} x$ is $|x| \geq 1$. For any $x$, $\sin x +2$ is positive, and a positive number can be raised to any power. (Recall negative numbers cannot be raised to any power–for example, $(-1)^{1/2}=\sqrt{-1}$ is not a real number.) So, the domain of $f(x)$ is $|x| \geq 1$.