Chain Rule with Several Independent Parameters

The formulas for finding derivatives of a composition of functions are called chain rules. The general case is when the variables of a several-variable (continuous) function are themselves (continuous) functions of several variables. These rules and implicit differentiation are detailed.

Proposition (Chain Rule with Several Independent Parameters) If is a differentiable function of the variables which in turn are differentiable functions of parameters then the composite function is differentiable and

Example (Chain Rule with Several Independent Parameters)

(a) Write out the chain rule for the case for the case when and   where and

    Solution. The chain rule for the case when and yields the following the partial derivatives:


and

(b)  If where and find the value of when and

    Solution. By the chain rule,

  When and we have and so


(c) If is differentiable and , show that

    Solution. Let , so Then according to the chain rule,

and

so that



(d) Find if , where and

    Solution. We have,

chain rule with several independent parameters _gr_50.gif]

(e) If is differentiable where and then find



    Solution. We compute,

Similarly,

and





so

(f) The Cauchy-Riemann equations are

where and Show that if and are expressed in terms of polar coordinates, the Cauchy-Riemann equations become



    Solution. Using and we can state the chain rule to be used:

By the chain rule,

and

Substituting,

we obtain

and so



Similarly, the chain rule to be used:

By the chain rule,

and

Substituting,

we obtain

and also

(g) If where and show that

    Solution. By the chain rule, we have

and

Therefore,

chain rule with several independent parameters _gr_96.gif]

and

Also,

Finally,

(h) If where show that

    Solution. By the chain rule, we have

and

Therefore,

and



Also,

Finally,

chain rule with several independent parameters _gr_122.gif]