# Math Help - differentiation of sinh function

1. ## differentiation of sinh function

I'm trying to differentiate sinh^(-1) (KL). (so arsinh(KL)) with respect to K and also with respect to L. Can some one point me in the right direction? Thanks

2. Hello, willowtree!

Differentiate: . $y \:=\:\sinh^{-1}(kx)$ with respect to $x.$
Take $\sinh$ of both sides: . $\sinh (y) \;=\;kx$

Differentiate implicitly: . $\cosh(y)\,\frac{dy}{dx} \:=\:k \quad\Rightarrow\quad \frac{dy}{dx} \:=\:\frac{k}{\cosh(y)}$

We have: . $\cosh^2(y) - \sinh^2(y) \:=\:1 \quad\Rightarrow\quad \cos^2(y) \:=\:1 + \sin^2(y) \;=\;1 + (kx)^2$

. . Hence: . $\cos(y) \:=\:\sqrt{1+k^2x^2}$

Therefore: . $\frac{dy}{dx} \;=\;\frac{k}{\sqrt{1+k^2x^2}}$