SOLUTION
The given function is:
[tex]f(x)=x^{6x}[/tex]
Rewrite the function in terms of y:
[tex]y=x^{6x}[/tex]
The derivative using logarithmic differentiation is done as follows:
[tex]\begin{gathered} \ln y=\ln x^{6x} \\ \ln y=6x\ln x \end{gathered}[/tex]
Differentiate both sides:
[tex]\frac{y^{\prime}}{y}=6x(\frac{1}{x})+6\ln x[/tex]
This is simplified to give:
[tex]y^{\prime}=y(6+6\ln x)[/tex]
Substituting the value of y into the equation gives:
[tex]y^{\prime}=x^{6x}(6+6\ln x)[/tex]
Therefore the derivative is:
[tex]f^{\prime}(x)=6x^{6x}(\ln x+1)[/tex]
The value of f'(1) is:
[tex]\begin{gathered} f^{\prime}(1)=6(1)^{6(1)}(\ln1+1) \\ f^{\prime}(1)=6(1)(0+1) \\ f^{\prime}(1)=6 \end{gathered}[/tex]
Therefore it follows f'(x)=6
