Let $f(x)=x^{7 x}$. Use logarithmic differentiation to determine the derivative. \[ f^{\prime}(x)= \] \[ f^{\prime}(1)= \]

Step 1 ：Let \(f(x)=x^{7 x}\). We want to find the derivative of this function.

Step 2 ：First, take the natural logarithm of both sides of the equation to get \(\ln(f) = 7x \ln(x)\).

Step 3 ：Differentiate both sides of the equation with respect to x to get \(\frac{f'}{f} = 7 \ln(x) + 7\).

Step 4 ：Solve for \(f'(x)\) by multiplying both sides of the equation by \(f(x)\) to get \(f'(x) = x^{7x+7}e^{7}\).

Step 5 ：Finally, substitute \(x = 1\) into the derivative to find the value of the derivative at \(x = 1\), which is \(f'(1) = e^{7}\).

Step 6 ：\(\boxed{f'(x) = x^{7x+7}e^{7}}\) is the derivative of the function \(f(x)=x^{7 x}\).

Step 7 ：\(\boxed{f'(1) = e^{7}}\) is the value of the derivative at \(x = 1\).