Question

To determine the derivative of the function \(f(x) = x\sinh x - \cosh x\).

Short answer

The derivative of the function \(f(x) = x\sinh x - \cosh x\) is \(x\cosh x\).

Step-by-step solution

Given Information

The equation to be proved is \(\frac{d}{{dx}}x\sinh x - \cosh x = x\cosh x\).

Formula of hyperbolic function

The derivatives of\(\sinh x\)is\(\frac{d}{{dx}}(\sin hx) = \cosh x\).

The derivatives of\(\cosh x\)is\(\frac{d}{{dx}}(\cos hx) = \sinh x\).

Derivation of \(\frac{d}{{dx}}x\sinh x - \cosh x = x\cosh x\)

Evaluate the derivative of\(f(x) = x\sinh x - \cosh x\).

\(\begin{aligned}{}{f^\prime }(x) &= {(x\sinh x - \cosh x)^\prime }\\ &= x(\cosh x) + (1)\sinh x - \sinh x\\ &= x\cosh x + \sinh x - \sinh x\\ &= x\cosh x\end{aligned}\)

Therefore, the derivative of the function \(f(x) = x\sinh x - \cosh x\) is \(x\cosh x\).