# What's the second derivative of ln(f(x))?

Oct 12, 2016

$\frac{f \left(x\right) f ' ' \left(x\right) - {\left(f ' \left(x\right)\right)}^{2}}{f \left(x\right)} ^ 2$

#### Explanation:

$y = \ln \left(f \left(x\right)\right)$

Through the chain rule and the knowledge that $\frac{d}{\mathrm{dx}} \ln \left(x\right) = \frac{1}{x}$, we see that:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{f} \left(x\right) \cdot f ' \left(x\right) = \frac{f ' \left(x\right)}{f} \left(x\right)$

To differentiate this again, we will use the quotient rule.

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = \frac{f \left(x\right) \frac{d}{\mathrm{dx}} f ' \left(x\right) - f ' \left(x\right) \frac{d}{\mathrm{dx}} f \left(x\right)}{f \left(x\right)} ^ 2$

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = \frac{f \left(x\right) f ' ' \left(x\right) - {\left(f ' \left(x\right)\right)}^{2}}{f \left(x\right)} ^ 2$