Higher-order derivatives

Exercise

1.

Show that the following formula．

(a)

$${(\sin{x})^{(n)} = \sin{(x + \frac{n \pi}{2})}}$$

(b)

$${(\cos{x})^{(n)} = \cos{(x + \frac{n \pi}{2})}}$$

(c)

$${\left[(1 + x)^{\alpha}\right]^{(n)} = \alpha(\alpha -1)\cdots(\alpha - n + 1)(1+x)^{\alpha - n}}$$

2.

Find the $n$th derivative of the following function．

(a)

$${f(x) = \frac{x^{3}}{1 - x}}$$

(b)

$${f(x) = x^{2} \sin{x}}$$

(c)

$${f(x) = e^{x} \sin{x}}$$