let (f(x)=x^{5x}). use logarithmic differentiation to determine the derivative (f(x)=square)

Explanation:

Step1: Take natural - log of both sides

Let (y = f(x)=x^{5x}). Then (\ln y=\ln(x^{5x})). Using the property of logarithms (\ln(a^b)=b\ln(a)), we have (\ln y = 5x\ln x).

Step2: Differentiate both sides with respect to (x)

The derivative of the left - hand side is (\frac{1}{y}y') by the chain rule. The derivative of the right - hand side using the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = 5x) and (v=\ln x). (u^\prime=5) and (v^\prime=\frac{1}{x}). So ((5x\ln x)^\prime=5\ln x + 5x\cdot\frac{1}{x}=5\ln x + 5). Thus, (\frac{1}{y}y'=5\ln x + 5).

Step3: Solve for (y')

Multiply both sides by (y). Since (y = x^{5x}), we get (y'=x^{5x}(5\ln x + 5)).

Answer:

(x^{5x}(5\ln x + 5))