Respuesta :
The fifth derivative of [tex]h(x)[/tex] is [tex]h^{(5)}(x) = (243\cdot e^{3\cdot x})\cdot (\cos 4x +9) + 5\cdot (81\cdot e^{3\cdot x})\cdot (-4\cdot \sin 4x)+10\cdot (27\cdot e^{3\cdot x})\cdot (-16\cdot \cos 4x)+10\cdot (9\cdot e^{3\cdot x})\cdot (64\cdot \sin 4x)+5\cdot (3\cdot e^{3\cdot x})\cdot (256\cdot \cos 4x) + (e^{3\cdot x}-8)\cdot (-1024\cdot \sin 4x)[/tex]. [tex]\blacksquare[/tex]
How to find the derivative of a product by Leibnitz rule
The Leibnitz rule states that the [tex]n[/tex]-th derivative of the product of two functions equals:
[tex](f\cdot g)^{(n)} = \Sigma\limits_{k=0}^{n}\left(\begin{array}{cc}n\\k\end{array} \right) f^{(n-k)}\cdot g^{(k)}[/tex] (1)
If we know that [tex]n = 5[/tex], [tex]f(x) = e^{3\cdot x}-8[/tex] and [tex]g(x) = \cos 4x + 9[/tex], then the derivative of [tex]h(x)[/tex] is:
Function f
[tex]f(x) = e^{3\cdot x}-8[/tex], [tex]f'(x) = 3\cdot e^{3\cdot x}[/tex], [tex]f''(x) = 9\cdot e^{3\cdot x}[/tex], [tex]f'''(x)= 27\cdot e^{3\cdot x}[/tex], [tex]f^{(4)}(x) = 81\cdot e^{3\cdot x}[/tex], [tex]f^{(5)}(x) = 243\cdot e^{3\cdot x}[/tex]
Function g
[tex]g(x) =\cos 4x +9[/tex], [tex]g'(x) = -4\cdot \sin 4x[/tex],[tex]g''(x) = -16\cdot \cos 4x[/tex], [tex]g'''(x) = 64\cdot \sin 4x[/tex],[tex]g^{(4)}(x) = 256\cdot \cos 4x[/tex], [tex]g^{(5)}(x) = -1024\cdot \sin 4x[/tex]
[tex]h^{(5)}(x) = \left(\begin{array}{cc}5\\0\end{array} \right)\cdot f^{(5)}(x)\cdot g(x) + \left(\begin{array}{cc}5\\1\end{array} \right)\cdot f^{(4)}(x)\cdot g'(x)+\left(\begin{array}{cc}5\\2\end{array} \right)\cdot f^{(3)}(x)\cdot g''(x)+\left(\begin{array}{cc}5\\3\end{array} \right)\cdot f^{(2)}(x)\cdot g'''(x)+\left(\begin{array}{cc}5\\4\end{array} \right)\cdot f(x)\cdot g^{(4)}(x)+\left(\begin{array}{cc}5\\5\end{array} \right)\cdot f(x)\cdot g^{(5)}(x)[/tex]
[tex]h^{(5)}(x) = f^{(5)}(x)\cdot g(x) + 5\cdot f^{(4)}(x)\cdot g'(x)+10\cdot f'''(x)\cdot g''(x)+10\cdot f''(x)\cdot g'''(x)+5\cdot f'(x)\cdot g^{(4)}(x)+f(x)\cdot g^{(5)}(x)[/tex]
[tex]h^{(5)}(x) = (243\cdot e^{3\cdot x})\cdot (\cos 4x +9) + 5\cdot (81\cdot e^{3\cdot x})\cdot (-4\cdot \sin 4x)+10\cdot (27\cdot e^{3\cdot x})\cdot (-16\cdot \cos 4x)+10\cdot (9\cdot e^{3\cdot x})\cdot (64\cdot \sin 4x)+5\cdot (3\cdot e^{3\cdot x})\cdot (256\cdot \cos 4x) + (e^{3\cdot x}-8)\cdot (-1024\cdot \sin 4x)[/tex]
The fifth derivative of [tex]h(x)[/tex] is [tex]h^{(5)}(x) = (243\cdot e^{3\cdot x})\cdot (\cos 4x +9) + 5\cdot (81\cdot e^{3\cdot x})\cdot (-4\cdot \sin 4x)+10\cdot (27\cdot e^{3\cdot x})\cdot (-16\cdot \cos 4x)+10\cdot (9\cdot e^{3\cdot x})\cdot (64\cdot \sin 4x)+5\cdot (3\cdot e^{3\cdot x})\cdot (256\cdot \cos 4x) + (e^{3\cdot x}-8)\cdot (-1024\cdot \sin 4x)[/tex]. [tex]\blacksquare[/tex]
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