Differentiate. \[ F(x)=\frac{e^{5 x}}{x^{15}} \] \[ F^{\prime}(x)= \]

Step 1 ：Given the function \(F(x)=\frac{e^{5 x}}{x^{15}}\)

Step 2 ：We need to find the derivative of this function. To do this, we will use the quotient rule, which states that the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Step 3 ：In this case, the numerator is \(e^{5x}\) and the denominator is \(x^{15}\). The derivative of \(e^{5x}\) is \(5e^{5x}\) and the derivative of \(x^{15}\) is \(15x^{14}\).

Step 4 ：Substituting these values into the quotient rule, we get \(F^{\prime}(x)=\frac{5e^{5x}}{x^{15}} - \frac{15e^{5x}}{x^{16}}\)

Step 5 ：Final Answer: \(F^{\prime}(x)=\boxed{\frac{5e^{5x}}{x^{15}} - \frac{15e^{5x}}{x^{16}}}\)