What are the first, second, and third order partial derivatives of f(x,y,z)=ln(xyx)?

Dec 30, 2015

Supposing it is $f \left(x , y , z\right) = \ln \left(x y z\right)$...

Explanation:

• First order derivatives:
$\frac{\delta f \left(x , y , z\right)}{\delta x} = \frac{1}{x y z} \cdot y z = \frac{1}{x}$
$\frac{\delta f \left(x , y , z\right)}{\delta y} = \frac{1}{x y z} \cdot x z = \frac{1}{y}$
$\frac{\delta f \left(x , y , z\right)}{\delta z} = \frac{1}{x y z} \cdot x y = \frac{1}{z}$

• Second order derivatives:
$\frac{\delta f {\left(x , y , z\right)}^{2}}{{\delta}^{2} x} = - \frac{1}{x} ^ 2$
$\frac{\delta f {\left(x , y , z\right)}^{2}}{{\delta}^{2} y} = - \frac{1}{y} ^ 2$
$\frac{\delta f {\left(x , y , z\right)}^{2}}{{\delta}^{2} z} = - \frac{1}{z} ^ 2$

• Third order derivatives:
$\frac{\delta f {\left(x , y , z\right)}^{3}}{{\delta}^{3} x} = \frac{2}{x} ^ 3$
$\frac{\delta f {\left(x , y , z\right)}^{3}}{{\delta}^{3} y} = \frac{2}{y} ^ 3$
$\frac{\delta f {\left(x , y , z\right)}^{3}}{{\delta}^{3} z} = \frac{2}{z} ^ 3$

BUT
If
your function is actually $f \left(x , y , z\right) = \ln \left(x y x\right) = \ln \left({x}^{2} y\right)$, then...

• First order derivatives:
$\frac{\delta f \left(x , y , z\right)}{\delta x} = \frac{1}{{x}^{2} y} \cdot 2 x y = \frac{1}{x}$
$\frac{\delta f \left(x , y , z\right)}{\delta y} = \frac{1}{{x}^{2} y} \cdot {x}^{2} = \frac{1}{y}$
$\frac{\delta f \left(x , y , z\right)}{\delta z} = 0$

• Second order derivatives:
$\frac{\delta f {\left(x , y , z\right)}^{2}}{{\delta}^{2} x} = - \frac{1}{x} ^ 2$
$\frac{\delta f {\left(x , y , z\right)}^{2}}{{\delta}^{2} y} = - \frac{1}{y} ^ 2$
$\frac{\delta f {\left(x , y , z\right)}^{2}}{{\delta}^{2} z} = 0$

• Third order derivatives:
$\frac{\delta f {\left(x , y , z\right)}^{3}}{{\delta}^{3} x} = \frac{2}{x} ^ 3$
$\frac{\delta f {\left(x , y , z\right)}^{3}}{{\delta}^{3} y} = \frac{2}{y} ^ 3$
$\frac{\delta f {\left(x , y , z\right)}^{3}}{{\delta}^{3} z} = 0$