Teorema 6.4.1. (Propriedade da derivada da transformada) Se $F\left(s\right)=\mathcal{L}\left\{f\left(t\right)\right\}$, então

Exemplo 6.4.1. Para calcular $\mathcal{L}\left\{t\cos <!--FUNCTION\; APPLICATION-->\left(wt\right)\right\}$, usamos a propriedade 6.4.1: $$\begin{array}{rcll}\mathcal{L}\left\{t\cos <!--FUNCTION\; APPLICATION-->\left(wt\right)\right\}& =& -\frac{d}{ds}\mathcal{L}\left\{\cos <!--FUNCTION\; APPLICATION-->\left(wt\right)\right\}& \\ & =& -\frac{d}{ds}\left(\frac{s}{s^2+w^2}\right)& \\ & =& -\frac{-s^2+w^2}{{\left(s^2+w^2\right)}^2}& \\ & =& \frac{s^2-w^2}{{\left(s^2+w^2\right)}^2}& \\ & & & \end{array}$$

E 6.4.1. Calcule $\mathcal{L}\left\{t^2\mathrm{sen}<!--FUNCTION\; APPLICATION-->\left(wt\right)\right\}$ usando a propriedade 6.4.1.

E 6.4.2. Use a propriedade da derivada da transformada para calcular $\mathcal{L}\left\{t\cos <!--FUNCTION\; APPLICATION-->\left(\omega t\right)\right\}$. A partir desta, prove que

E 6.4.3. Usando a derivada da transformada de Laplace, calcule:

• $\mathcal{L}\left\{t\mathrm{senh}<!--FUNCTION\; APPLICATION-->\left(at\right)\right\}$
• $\mathcal{L}\left\{t\cosh <!--FUNCTION\; APPLICATION-->\left(at\right)\right\}$