4.5 The chain rule  (Page 5/9)

[link] is a direct consequence of [link] . In particular, if we assume that $y$ is defined implicitly as a function of $x$ via the equation $f\left(x,y\right)=0,$ we can apply the chain rule to find $dy\text{/}dx\text{:}$

Solving this equation for $dy\text{/}dx$ gives [link] . [link] can be derived in a similar fashion.

Let’s now return to the problem that we started before the previous theorem. Using [link] and the function $f\left(x,y\right)=x^2+3y^2+4y-4,$ we obtain

Then [link] gives

which is the same result obtained by the earlier use of implicit differentiation.

Key concepts

• The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.
• Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables.

Key equations

• Chain rule, one independent variable
  $\frac{dz}{dt}=\frac{\partial z}{\partial x}·\frac{dx}{dt}+\frac{\partial z}{\partial y}·\frac{dy}{dt}$
• Chain rule, two independent variables
  $\frac{dz}{du}=\frac{\partial z}{\partial x}·\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}·\frac{\partial y}{\partial u}$
  $\frac{dz}{dv}=\frac{\partial z}{\partial x}·\frac{\partial x}{\partial v}+\frac{\partial z}{\partial y}·\frac{\partial y}{\partial v}$
• Generalized chain rule
  $\frac{\partial w}{\partial t_j}=\frac{\partial w}{\partial x_1}\frac{\partial x_1}{\partial t_j}+\frac{\partial w}{\partial x_2}\frac{\partial x_1}{\partial t_j}+\text{⋯}+\frac{\partial w}{\partial x_m}\frac{\partial x_m}{\partial t_j}$

For the following exercises, use the information provided to solve the problem.