Question

differentiate ( f(x)=x cosh x+3 sinh x ).

( 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:

\(x\sinh x + 4\cosh x\)