Chapter 12, Problem 21P

(a)

To determine

Prove that the expression $\left[-\frac{dF\left(s\right)}{ds}=L\left\{tf\left(t\right)\right\}\right]$.

(b)

To determine

Prove that the expression ${\left(-1\right)}^n\frac{d^{n}F\left(s\right)}{d{s}^n}=L\left\{t^{n}f\left(t\right)\right\}$.

(c)

To determine

Find the Laplace transform of the given functions using the result of Part (b).