Question:

<b>Standard Form — Inverse Hyperbolic</b><br>This is the inverse sinh form: $y = \sinh^{-1}\\!\left(\frac{x}{a}\right)$.<br>Direct differentiation: $\frac{dy}{dx} = \frac{d}{dx}\left[\ln\\!\left(x+\sqrt{a^2+x^2}\right) - \ln a\right]$<br>$= \frac{1}{x+\sqrt{a^2+x^2}} \cdot \left(1 + \frac{x}{\sqrt{a^2+x^2}}\right) = \frac{1}{\sqrt{a^2+x^2}}$<br><b>Key concept:</b> $\frac{d}{dx}\sinh^{-1}\\!\left(\frac{x}{a}\right) = \frac{1}{\sqrt{a^2+x^2}}$<br><b>Trap:</b> Students compute the full quotient derivative unnecessarily; recognise the standard form.