# What is the second derivative of f(x)= ln sqrt(e^x)?

Jan 15, 2016

$0$

#### Explanation:

Let $t = \sqrt{{e}^{x}} = {e}^{\frac{x}{2}}$

$\frac{\mathrm{dt}}{\mathrm{dx}} = \frac{1}{2} \cdot {e}^{\frac{x}{2}}$

$f \left(t\right) = \ln t$

$\frac{\mathrm{df}}{\mathrm{dt}} = \frac{1}{t}$

$\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{df}}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}$

$= \frac{1}{t} \cdot \frac{{e}^{\frac{x}{2}}}{2} = \frac{{e}^{\frac{x}{2}}}{2 {e}^{\frac{x}{2}}} = \frac{1}{2}$

$f ' ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(\frac{1}{2}\right) = 0$