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

To differentiate the function \( f(x) = x^{9} e^{8x} \), we'll use the product rule, which states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative \( (uv)' = u'v + uv' \). Here, let \( u = x^{9} \) and \( v = e^{8x} \). First, find \( u' = 9x^{8} \) and \( v' = 8e^{8x} \). Now apply the product rule: \[ f'(x) = u'v + uv' = (9x^{8})(e^{8x}) + (x^{9})(8e^{8x}) = 9x^{8} e^{8x} + 8x^{9} e^{8x}. \] Combining these gives: \[ f'(x) = e^{8x}(9x^{8} + 8x^{9}). \] Thus, the derivative is \( f^{\prime}(x) = e^{8x}(8x^{9} + 9x^{8}) \). Now go grab a snack as you dive into the world of derivatives – it can be quite the adventure! Just remember, when using product and chain rules, double-check your derivatives to avoid silly mistakes, like forgetting to multiply by the derivative of the exponent in an exponential function. Happy differentiating!