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

Answer

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Answer

$f^{'} (1) = e^{7}$ is the value of the derivative at $x = 1$ .

Steps

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 $\ln (f) = 7 x \ln (x)$ .

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 $f^{'} (x) = x^{7 x + 7} e^{7}$ .

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 ： $f^{'} (x) = x^{7 x + 7} e^{7}$ is the derivative of the function $f (x) = x^{7 x}$ .

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