- How do you write each of the following as an algebraic function of \(x\)? \[ \sin\left(\arctan(x)\right) \qquad \qquad \sec\left(\arcsin(x)\right) \]
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Demonstrate how to calculate these algebraic formulas for the derivatives of inverse trigonometric functions.
\[ \frac{\mathrm{d}}{\mathrm{d}x} \arcsin(x) = \frac{1}{\sqrt{1-x^2}} \qquad \frac{\mathrm{d}}{\mathrm{d}x} \mathrm{arccsc}(x) = -\frac{1}{x\sqrt{x^2-1}} \] \[ \frac{\mathrm{d}}{\mathrm{d}x} \arccos(x) = -\frac{1}{\sqrt{1-x^2}} \qquad \frac{\mathrm{d}}{\mathrm{d}x} \mathrm{arcsec}(x) = \frac{1}{x\sqrt{x^2-1}} \] \[ \frac{\mathrm{d}}{\mathrm{d}x} \arctan(x) = \frac{1}{1+x^2} \qquad \frac{\mathrm{d}}{\mathrm{d}x} \mathrm{arccot}(x) = -\frac{1}{1+x^2} \]