Find the derivative of the given function. \[ y=27^{5 x} \]

Simplify the expression by following steps: - step0: Solution: \(\frac{d}{dx\times 27^{5x}}\) - step1: Reduce the fraction: \(\frac{1}{x\times 27^{5x}}\) To find the derivative of the function \[ y = 27^{5x} \] we can use the property of exponential functions. The derivative of an exponential function \(a^{u}\) with respect to \(x\) is given by: \[ \frac{d}{dx}(a^{u}) = a^{u} \cdot \ln(a) \cdot \frac{du}{dx} \] In our case, \(a = 27\) and \(u = 5x\). 1. First, we find \(\frac{du}{dx}\): \[ \frac{du}{dx} = \frac{d}{dx}(5x) = 5 \] 2. Now, we can apply the derivative formula: \[ \frac{dy}{dx} = 27^{5x} \cdot \ln(27) \cdot 5 \] Thus, the derivative of the function is: \[ \frac{dy}{dx} = 5 \cdot 27^{5x} \cdot \ln(27) \] This is the final result for the derivative of the given function.