# How do you use chain rule to find partial derivatives?

##### 1 Answer
Apr 1, 2015

One type of example is as follows: If $z = f \left(x , y\right)$, $x = g \left(t\right)$, and $y = h \left(t\right)$, then you can write $\setminus \frac{\mathrm{dz}}{\mathrm{dt}} = \setminus \frac{\setminus \partial z}{\setminus \partial x} \setminus \frac{\mathrm{dx}}{\mathrm{dt}} + \setminus \frac{\setminus \partial z}{\setminus \partial y} \setminus \frac{\mathrm{dy}}{\mathrm{dt}}$.

For example, if $z = {x}^{2} + {y}^{3}$, $x = {t}^{4}$, and $y = {t}^{5}$, then $\setminus \frac{\setminus \partial z}{\setminus \partial t} = 2 x \setminus \cdot 4 {t}^{3} + 3 {y}^{2} \setminus \cdot 5 {t}^{4} = 8 {t}^{4} \setminus \cdot {t}^{3} + 15 {t}^{10} \setminus \cdot {t}^{4} = 8 {t}^{7} + 15 {t}^{14.}$

You can check that this works by noting that $z = {x}^{2} + {y}^{3} = {t}^{8} + {t}^{15}$ so that $8 {t}^{7} + 15 {t}^{14}$ (so the Chain Rule is more work for this example).

There are other examples where the Chain Rule can be less work.