Second derivative of $$$\operatorname{asec}{\left(x \right)}$$$

The calculator will find the second derivative of $$$\operatorname{asec}{\left(x \right)}$$$, with steps shown.

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Find $$$\frac{d^{2}}{dx^{2}} \left(\operatorname{asec}{\left(x \right)}\right)$$$.

Solution

Find the first derivative $$$\frac{d}{dx} \left(\operatorname{asec}{\left(x \right)}\right)$$$

The derivative of the inverse secant is $$$\frac{d}{dx} \left(\operatorname{asec}{\left(x \right)}\right) = \frac{1}{x^{2} \sqrt{1 - \frac{1}{x^{2}}}}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\operatorname{asec}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{1}{x^{2} \sqrt{1 - \frac{1}{x^{2}}}}\right)}$$

Simplify:

$$\frac{1}{x^{2} \sqrt{1 - \frac{1}{x^{2}}}} = \frac{\left|{x}\right|}{x^{2} \sqrt{x^{2} - 1}}$$

Thus, $$$\frac{d}{dx} \left(\operatorname{asec}{\left(x \right)}\right) = \frac{\left|{x}\right|}{x^{2} \sqrt{x^{2} - 1}}$$$.

Next, $$$\frac{d^{2}}{dx^{2}} \left(\operatorname{asec}{\left(x \right)}\right) = \frac{d}{dx} \left(\frac{\left|{x}\right|}{x^{2} \sqrt{x^{2} - 1}}\right)$$$

Apply the quotient rule $$$\frac{d}{dx} \left(\frac{f{\left(x \right)}}{g{\left(x \right)}}\right) = \frac{\frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} - f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)}{g^{2}{\left(x \right)}}$$$ with $$$f{\left(x \right)} = \left|{x}\right|$$$ and $$$g{\left(x \right)} = x^{2} \sqrt{x^{2} - 1}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\frac{\left|{x}\right|}{x^{2} \sqrt{x^{2} - 1}}\right)\right)} = {\color{red}\left(\frac{\frac{d}{dx} \left(\left|{x}\right|\right) x^{2} \sqrt{x^{2} - 1} - \left|{x}\right| \frac{d}{dx} \left(x^{2} \sqrt{x^{2} - 1}\right)}{\left(x^{2} \sqrt{x^{2} - 1}\right)^{2}}\right)}$$

Apply the product rule $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ with $$$f{\left(x \right)} = x^{2}$$$ and $$$g{\left(x \right)} = \sqrt{x^{2} - 1}$$$:

$$\frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left|{x}\right| {\color{red}\left(\frac{d}{dx} \left(x^{2} \sqrt{x^{2} - 1}\right)\right)}}{x^{4} \left(x^{2} - 1\right)} = \frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left|{x}\right| {\color{red}\left(\frac{d}{dx} \left(x^{2}\right) \sqrt{x^{2} - 1} + x^{2} \frac{d}{dx} \left(\sqrt{x^{2} - 1}\right)\right)}}{x^{4} \left(x^{2} - 1\right)}$$

The function $$$\sqrt{x^{2} - 1}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \sqrt{u}$$$ and $$$g{\left(x \right)} = x^{2} - 1$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$$\frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(x^{2} {\color{red}\left(\frac{d}{dx} \left(\sqrt{x^{2} - 1}\right)\right)} + \sqrt{x^{2} - 1} \frac{d}{dx} \left(x^{2}\right)\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)} = \frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(x^{2} {\color{red}\left(\frac{d}{du} \left(\sqrt{u}\right) \frac{d}{dx} \left(x^{2} - 1\right)\right)} + \sqrt{x^{2} - 1} \frac{d}{dx} \left(x^{2}\right)\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)}$$

Apply the power rule $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ with $$$n = \frac{1}{2}$$$:

$$\frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(x^{2} {\color{red}\left(\frac{d}{du} \left(\sqrt{u}\right)\right)} \frac{d}{dx} \left(x^{2} - 1\right) + \sqrt{x^{2} - 1} \frac{d}{dx} \left(x^{2}\right)\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)} = \frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(x^{2} {\color{red}\left(\frac{1}{2 \sqrt{u}}\right)} \frac{d}{dx} \left(x^{2} - 1\right) + \sqrt{x^{2} - 1} \frac{d}{dx} \left(x^{2}\right)\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)}$$

Return to the old variable:

$$\frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(\frac{x^{2} \frac{d}{dx} \left(x^{2} - 1\right)}{2 \sqrt{{\color{red}\left(u\right)}}} + \sqrt{x^{2} - 1} \frac{d}{dx} \left(x^{2}\right)\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)} = \frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(\frac{x^{2} \frac{d}{dx} \left(x^{2} - 1\right)}{2 \sqrt{{\color{red}\left(x^{2} - 1\right)}}} + \sqrt{x^{2} - 1} \frac{d}{dx} \left(x^{2}\right)\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)}$$

The derivative of a sum/difference is the sum/difference of derivatives:

$$\frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(\frac{x^{2} {\color{red}\left(\frac{d}{dx} \left(x^{2} - 1\right)\right)}}{2 \sqrt{x^{2} - 1}} + \sqrt{x^{2} - 1} \frac{d}{dx} \left(x^{2}\right)\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)} = \frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(\frac{x^{2} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right) - \frac{d}{dx} \left(1\right)\right)}}{2 \sqrt{x^{2} - 1}} + \sqrt{x^{2} - 1} \frac{d}{dx} \left(x^{2}\right)\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:

$$\frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(\frac{x^{2} \left({\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} - \frac{d}{dx} \left(1\right)\right)}{2 \sqrt{x^{2} - 1}} + \sqrt{x^{2} - 1} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)}\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)} = \frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(\frac{x^{2} \left({\color{red}\left(2 x\right)} - \frac{d}{dx} \left(1\right)\right)}{2 \sqrt{x^{2} - 1}} + \sqrt{x^{2} - 1} {\color{red}\left(2 x\right)}\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)}$$

The derivative of a constant is $$$0$$$:

$$\frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(\frac{x^{2} \left(2 x - {\color{red}\left(\frac{d}{dx} \left(1\right)\right)}\right)}{2 \sqrt{x^{2} - 1}} + 2 x \sqrt{x^{2} - 1}\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)} = \frac{x^{2} \sqrt{x^{2} - 1} \frac{d}{dx} \left(\left|{x}\right|\right) - \left(\frac{x^{2} \left(2 x - {\color{red}\left(0\right)}\right)}{2 \sqrt{x^{2} - 1}} + 2 x \sqrt{x^{2} - 1}\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)}$$

The derivative of the absolute value is $$$\frac{d}{dx} \left(\left|{x}\right|\right) = \frac{x}{\left|{x}\right|}$$$:

$$\frac{x^{2} \sqrt{x^{2} - 1} {\color{red}\left(\frac{d}{dx} \left(\left|{x}\right|\right)\right)} - \left(\frac{x^{3}}{\sqrt{x^{2} - 1}} + 2 x \sqrt{x^{2} - 1}\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)} = \frac{x^{2} \sqrt{x^{2} - 1} {\color{red}\left(\frac{x}{\left|{x}\right|}\right)} - \left(\frac{x^{3}}{\sqrt{x^{2} - 1}} + 2 x \sqrt{x^{2} - 1}\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)}$$

Simplify:

$$\frac{\frac{x^{3} \sqrt{x^{2} - 1}}{\left|{x}\right|} - \left(\frac{x^{3}}{\sqrt{x^{2} - 1}} + 2 x \sqrt{x^{2} - 1}\right) \left|{x}\right|}{x^{4} \left(x^{2} - 1\right)} = \frac{1 - 2 x^{2}}{x \left(x^{2} - 1\right)^{\frac{3}{2}} \left|{x}\right|}$$

Thus, $$$\frac{d}{dx} \left(\frac{\left|{x}\right|}{x^{2} \sqrt{x^{2} - 1}}\right) = \frac{1 - 2 x^{2}}{x \left(x^{2} - 1\right)^{\frac{3}{2}} \left|{x}\right|}$$$.

Therefore, $$$\frac{d^{2}}{dx^{2}} \left(\operatorname{asec}{\left(x \right)}\right) = \frac{1 - 2 x^{2}}{x \left(x^{2} - 1\right)^{\frac{3}{2}} \left|{x}\right|}$$$.

Answer

$$$\frac{d^{2}}{dx^{2}} \left(\operatorname{asec}{\left(x \right)}\right) = \frac{1 - 2 x^{2}}{x \left(x^{2} - 1\right)^{\frac{3}{2}} \left|{x}\right|}$$$A