Question

cosh x dy/dx+ ysinh^2x=2coshx sinhx​

Answer 1

Answer:

Proofs of Derivatives of Hyperbolics

Proof of (d/dx)sinh(x) = cosh(x) : From the derivative of ex

Given: sinh(x) = ( ex - e-x )/2; cosh(x) = (ex + e-x)/2; (d/dx) ( f(x)+g(x) ) =(d/dx) f(x) + (d/dx) g(x); Chain Rule; (d/dx)( c*f(x) ) = c (d/dx)f(x).

Solve:

(d/dx) sinh(x)= (d/dx) ( ex- e-x )/2 = 1/2 (d/dx)(ex) -1/2 (d/dx)(e-x)

= 1/2 ex + 1/2 e-x = ( ex + e-x )/2 = cosh(x) Q.E.D