Derivative of Arccotangent Function/Proof 1
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Theorem
- $\dfrac {\map \d {\arccot x} } {\d x} = \dfrac {-1} {1 + x^2}$
Proof
| \(\ds y\) | \(=\) | \(\ds \arccot x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \cot y\) | Definition of Real Arccotangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d y}\) | \(=\) | \(\ds -\csc^2 y\) | Derivative of Cotangent Function | ||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {1 + \cot^2 y}\) | Difference of Squares of Cosecant and Cotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {1 + x^2}\) | Definition of $x$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \frac {-1} {1 + x^2}\) | Derivative of Inverse Function |
$\blacksquare$