Take Derivative Of Logarithm with Coefficient

Question

IN my books, I see this example that derivative of $ln(\frac{1}{x^3})$ is $\frac{-3}{x}$. I know derivative of just $ln x$ is $\frac{1}{x}$. What is rule for multiplying coefficient with 1/x? I not see that in my book anywhere.

Answer 1

If you can't remember logarithm rules, you could use the chain rule:

$$\frac d {dx} ln\left(\frac{1}{x^3}\right)= \frac 1 {\frac 1 {x^3}} \frac d {dx} \frac 1 {x^3} = x^3 \times-3x^{-4} = -3/x.$$