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105-8b-ChainRule

# 105-8b-ChainRule - Week 8 The Chain Rule The Chain Rule for...

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Week 8 The Chain Rule The Chain Rule for z = f ( x ( t ) , y ( t )) In two variables, there are different kinds of Chain Rules; the one we will see is for situations like z = f ( x ( t ) , y ( t )), where z is a function of x and y , and x and y are both functions of t . In such a situation, t is called the independent variable , because it doesn’t depend on any other variable, while x , y and z are called dependent variables , since they depend on other variables. Recall that one way to write the 1-variable Chain Rule for y = f ( x ( t )) is dy dt = dy dx dx dt . The same notation works best for the 2-variable Chain Rule with 1 independent variable: dz dt = ∂z ∂x dx dt + ∂z ∂y dy dt Note that since z = f ( x, y ) is a 2-variable function, ∂z ∂x and ∂z ∂y are partial derivatives, so are written with ’s, while the other derivatives are of 1-variable functions, so are written with d ’s. This notation is nice and short, but can be a bit misleading; we should also write it out more explicitly, with z = g ( t ) denoting z directly as a function of t : z = g ( t ) = f ( x ( t ) , y ( t )) dz dt = g 0 ( t ) = f x ( x ( t ) , y ( t )) · x 0 ( t ) + f y ( x ( t ) , y ( t )) · y 0 ( t ) . Then if we plug in a value t = a , we get: dz dt t = a = g 0 ( a ) = f x ( x ( a ) , y ( a )) · x 0 ( a ) + f y ( x ( a ) , y ( a )) · y 0 ( a ) . Note that in theory we could always write out z = g ( t ) explicitly, and differentiate that as a 1-variable function; but in practice this might be too much work, since this will be a more complicated function, hence more work to differentiate.

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105-8b-ChainRule - Week 8 The Chain Rule The Chain Rule for...

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