Question

Тридцатая производная от функции f(x)=(sh(5x)) * (x^3) по формуле Лейбница

Answer 1

$[(\sinh(5x))\cdot(x^3)]^{(30)} = \sum\limits_{j=0}^{30}\binom{30}{j}(\sinh(5x))^{(30-j)}(x^{3})^{(j)} = \binom{30}{0}(\sinh(5x))^{(30)}(x^{3})^{(0)}+\binom{30}{1}(\sinh(5x))^{(29)}(x^{3})^{(1)}+\binom{30}{2}(\sinh(5x))^{(28)}(x^{3})^{(2)}+\binom{30}{3}(\sinh(5x))^{(27)}(x^{3})^{(3)} = \boxed{5^{30}x^{3}\sinh(5x)+90\cdot5^{29}x^{2}\cosh(5x)+3\cdot 30\cdot 29\cdot 5^{28}x\sinh(5x)+30\cdot 29\cdot 28\cdot 5^{27}\cosh(5x)}$