Differentiate the following function. \( f(x)=x^{9} e^{8 x} \) \( f^{\prime}(x)=\square \)

The function \( f(x) = x^{9} e^{8x} \) is not in a form that can be differentiated directly. To differentiate this function, we can use the product rule of differentiation, which states that if we have a function of the form \( f(x) = u(x)v(x) \), then the derivative of \( f(x) \) is given by \( f'(x) = u'(x)v(x) + u(x)v'(x) \). In this case, we can rewrite the function as \( f(x) = x^{9} \cdot e^{8x} \), where \( u(x) = x^{9} \) and \( v(x) = e^{8x} \). Now, we can find the derivatives of \( u(x) \) and \( v(x) \) separately: \( u'(x) = 9x^{8} \) \( v'(x) = 8e^{8x} \) Using the product rule, we can now find the derivative of \( f(x) \): \( f'(x) = u'(x)v(x) + u(x)v'(x) \) \( f'(x) = 9x^{8} \cdot e^{8x} + x^{9} \cdot 8e^{8x} \) \( f'(x) = 9x^{8}e^{8x} + 8x^{9}e^{8x} \) Therefore, the derivative of the function \( f(x) = x^{9} e^{8x} \) is \( f'(x) = 9x^{8}e^{8x} + 8x^{9}e^{8x} \).