How do you find the first and second derivative of ln(x^3)?

Oct 19, 2016

$\frac{d}{\mathrm{dx}} \ln \left({x}^{3}\right) = \frac{3}{x}$

$\frac{{d}^{2}}{{\mathrm{dx}}^{2}} \ln \left({x}^{3}\right) = - \frac{3}{x} ^ 2$

Explanation:

Using the chain rule, the power rule, and the product rule, along with the derivative $\frac{d}{\mathrm{dx}} \ln \left(x\right) = \frac{1}{x}$, we have

First Derivative:

$\frac{d}{\mathrm{dx}} \ln \left({x}^{3}\right) = \frac{1}{x} ^ 3 \left(\frac{d}{\mathrm{dx}} {x}^{3}\right)$

$= \frac{1}{x} ^ 3 \left(3 {x}^{2}\right)$

$= \frac{3}{x}$

Second Derivative:

$\frac{{d}^{2}}{{\mathrm{dx}}^{2}} \ln \left({x}^{3}\right) = \frac{d}{\mathrm{dx}} \left(\frac{d}{\mathrm{dx}} \ln \left({x}^{3}\right)\right)$

$= \frac{d}{\mathrm{dx}} \left(\frac{3}{x}\right)$

$= \frac{d}{\mathrm{dx}} 3 {x}^{- 1}$

$= 3 \left(- {x}^{-} 2\right)$

$= - \frac{3}{x} ^ 2$