Calculus III - Higher Order Partial Derivatives

Hide Mobile Notice

Mobile Notice

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 13.4 : Higher Order Partial Derivatives

Back to Problem List

7. Given $f\left( {x,y,z} \right) = {x^4}{y^3}{z^6}$ find $\displaystyle \frac{{{\partial ^6}f}}{{\partial y\partial {z^2}\partial y\partial {x^2}}}$ .

Show All Steps Hide All Steps

Start Solution

Through a natural extension of Clairaut’s theorem we know we can do these partial derivatives in any order we wish to. However, in this case there doesn’t seem to be any reason to do anything other than the order shown in the problem statement.

Here is the first derivative we need to take.

$\frac{{\partial f}}{{\partial x}} = 4{x^3}{y^3}{z^6}$ Show Step 2

The second derivative is,

$\frac{{{\partial ^2}f}}{{\partial {x^2}}} = 12{x^2}{y^3}{z^6}$ Show Step 3

The third derivative is,

$\frac{{{\partial ^3}f}}{{\partial y\partial {x^2}}} = 36{x^2}{y^2}{z^6}$ Show Step 4

The fourth derivative is,

$\frac{{{\partial ^4}f}}{{\partial z\partial y\partial {x^2}}} = 216{x^2}{y^2}{z^5}$ Show Step 5

The fifth derivative is,

$\frac{{{\partial ^5}f}}{{\partial {z^2}\partial y\partial {x^2}}} = 1080{x^2}{y^2}{z^4}$ Show Step 6

The sixth and final derivative we need for this problem is,

$\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{{\partial ^6}f}}{{\partial y\partial {z^2}\partial y\partial {x^2}}} = 2160{x^2}y{z^4}}}$