Logarithmic Differentiation — Question 5

Question 5 - Solution

The real domain is $x>0,\ x\ne4$. Where the function is nonzero, use absolute values in logarithmic differentiation:

$$\ln|f(x)|=5\ln(x^2+1)+\tfrac13\ln|\ln x|-x^3-2\ln|x-4|$$

Differentiating gives

$$\boxed{f'(x)=f(x)\left(\frac{10x}{x^2+1}+\frac1{3x\ln x}-3x^2-\frac2{x-4}\right).}$$

This formula holds for $x>0,\ x\ne1,4$. At $x=1$, $f(1)=0$, but the cube-root factor gives no finite derivative.