Logarithmic Differentiation — Question 2

Question 2

Let the function be: $$f(x) = \frac{(x^2 + 1)^5 \cdot \sqrt{x + 3}}{(x - 1)^4}$$

• Use logarithmic differentiation to find $f'(x)$.

• Simplify your final expression as much as possible.

Question 2 - Solution

The real domain is $x\geq-3,\ x\ne1$. Where the function is nonzero, use absolute values in logarithmic differentiation:

$$\ln|f(x)|=5\ln(x^2+1)+\tfrac12\ln(x+3)-4\ln|x-1|$$

Differentiating gives

$$\boxed{f'(x)=f(x)\left(\frac{10x}{x^2+1}+\frac1{2(x+3)}-\frac4{x-1}\right).}$$

This formula holds for $x>-3,\ x\ne1$. At $x=-3$ there is no finite derivative because of the square-root factor.