Use the convolution integral and the notation and results of Chapter Problem 5.6 to derive the following alternative expression for the response of a system subject to an excitation, \(F(t)\) :

\[ x(t)=F(0) g(t)+\int_{0}^{t} \frac{d F(\tau)}{d \tau} g(t-\tau) d \tau \]

Data From Chapter Problem 5.6:

Let \(g(t)\) be the response of an underdamped system to a unit step function and \(h(t)\) the response of an underdamped system to a unit impulse function. Show

\[ h(t)=\frac{d g}{d t} \]