differentiate ( f(x)=x cosh x+3 sinh x ).\n\n( f^{prime}(x)= )

Explanation:

Step1: Differentiate (x\cosh x) using product rule

The product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = x), (u^\prime=1), (v=\cosh x), (v^\prime=\sinh x). So ((x\cosh x)^\prime=1\times\cosh x+x\sinh x=\cosh x + x\sinh x)

Step2: Differentiate (3\sinh x)

Using the rule ((a\cdot f(x))^\prime=a\cdot f^\prime(x)) ((a = 3), (f(x)=\sinh x), (f^\prime(x)=\cosh x)), we get ((3\sinh x)^\prime=3\cosh x)

Step3: Sum the derivatives

(f^\prime(x)=(x\cosh x)^\prime+(3\sinh x)^\prime=\cosh x + x\sinh x+3\cosh x) Simplify: (f^\prime(x)=x\sinh x + 4\cosh x)

Answer:

(x\sinh x + 4\cosh x)