Derivative of Real Area Hyperbolic Cosine of Function

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $u$ be a differentiable real function of $x$.

Then:

$\map {\dfrac \d {\d x} } {\arcosh u} = \dfrac 1 {\sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$

where $u > 1$

where $\cosh^{-1}$ is the real area hyperbolic cosine.


Proof

\(\ds \map {\frac \d {\d x} } {\arcosh u}\) \(=\) \(\ds \map {\frac \d {\d u} } {\arcosh u} \frac {\d u} {\d x}\) Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac 1 {\sqrt {u^2 - 1} } \frac {\d u} {\d x}\) Derivative of Real Area Hyperbolic Cosine

$\blacksquare$


Also see


Sources