The upper branch corresponds to the variable x x and the lower branch corresponds to the variable y. Since f f has two independent variables, there are two lines coming from this corner.
In this diagram, the leftmost corner corresponds to z = f ( x, y ). This pattern works with functions of more than two variables as well, as we see later in this section.įigure 4.34 Tree diagram for the case d z d t = ∂ z ∂ x Two terms appear on the right-hand side of the formula, and f f is a function of two variables. The variables x and y x and y that disappear in this simplification are often called intermediate variables: they are independent variables for the function f, f, but are dependent variables for the variable t. If we treat these derivatives as fractions, then each product “simplifies” to something resembling ∂ f / d t. Recall that when multiplying fractions, cancelation can be used. The first term in the equation is ∂ f ∂ x This proves the chain rule at t = t 0 t = t 0 the rest of the theorem follows from the assumption that all functions are differentiable over their entire domains.Ĭloser examination of Equation 4.29 reveals an interesting pattern.
Since x ( t ) x ( t ) and y ( t ) y ( t ) are both differentiable functions of t, t, both limits inside the last radical exist.
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