Differentiating Hyperbolic Functions

Find \(y^\prime \).

1. \(y=x^{2}-2\sinh x\)
2. \(y=\cosh (x^{2}+1) \)

Solution

(a) \(y^\prime =\dfrac{d}{dx}( x^{2}-2\sinh x) =2x-2\dfrac{d}{dx}\sinh x=2x-2\cosh x\)

(b) We use the Chain Rule with \(y=\cosh u\) and \(u=x^{2}+1\). \[ y^\prime =\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}=\sinh u\cdot 2x=2x\sinh (x^{2}+1) \]