Arccot

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The function $\mathrm{arccot} \colon \mathbb{R} \rightarrow \left( - \dfrac{\pi}{2}, \dfrac{\pi}{2} \right] \setminus \{0\}$ is the inverse function of the cotangent function.

Properties

Theorem

The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{arccot}(z) = -\dfrac{1}{z^2+1},$$ where $\mathrm{arccot}$ denotes the inverse cotangent function.

Proof

If $y=\mathrm{arccot}(z)$ then $\cot(y)=z$. Now use implicit differentiation with respect to $z$ to get $$-\csc^2(y)y'=1.$$ Substituting back in $y=\mathrm{arccos}(z)$ yields the formula $$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{arccot}(z) = -\dfrac{1}{\csc^2(\mathrm{arccot}(z))} = -\dfrac{1}{z^2+1},$$ as was to be shown. █

References

References

Which is the correct graph of arccot x?

See Also

Cotangent
Coth
Arccoth

<center>Inverse trigonometric functions
</center>