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Multivariable Chain Rule (Example 1)
You use the multivariable chain rule when you have functions that are functions of other functions, basically when there are equations that look like they could be substituted into each other.
Let’s do our own practice example.
Find \(\frac{{dz}}{{dt}}\) using the chain rule, where \(z = {x^2} + {y^2} + xy\), \(x = sin (t)\) and \(y = {e^t}\).
See how you could technically substitute the equations into each other and then find dz/dt that way?
But it’s actually a lot easier to use the multivariable chain rule.
The best way to do it is map out how the variables are related to each other.
- z is a function of x and y
- x and y are functions of t
Then,
\[\frac{{dz}}{{dt}} = \frac{{dz}}{{dx}} \cdot \frac{{dx}}{{dt}} + \frac{{dz}}{{dy}} \cdot \frac{{dy}}{{dt}}\]
So find all these partial derivatives and substitute them in.
\[\frac{{dz}}{{dx}} = 2x + y\]
\[\frac{{dx}}{{dt}} = \cos (t)\]
\[\frac{{dz}}{{dy}} = 2y + x\]
\[\frac{{dx}}{{dt}} = {e^t}\]
Then,
\[\frac{{dz}}{{dt}} = (2x + y)\cos (t) + (2y + x){e^t}\]
Click here to try the next and harder multivarible chain rule example!