Find $y^\prime$.

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 $u=x^{2}+1$. $$y^\prime =\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}=\sinh u\cdot 2x=2x\sinh (x^{2}+1)$$