Chain rule of a function of two variables for one independent variable

 Goal: State the chain rule for one independent variable

Chain rule for one independent variable

In a function of one variable, one of the most useful rules of differentiation is the chain rule. The same concept is applied for the differentiation of functions of more than one variable.

Recall that the chain rule for a function of one variable is given by:

In this equation both f(x) and g(x) are functions of one variable. Let's suppose that f is a function of two variables and g a function of one variable, then the derivative of of the composition of the two functions is given by the following theorem:

Theorem

Let's suppose that x = g(t) and y = h(t) are two differentiable functions of t and z = f(x,y) a differentiable function of x and y, then z = (x(t), y(t)) is a differentiable function of t and

where the ordinary derivatives are evaluated at t and the partial derivatives at (x,y).

Example

Calculate dz/dt for each of the following functions:

Solution

a.  In order to use the chain rule, let's calculate:

We have:

Let's substitute these quantities in the formula that allows to calculate dz/dt

b. Let's calculate the 4quantities representing the partial derivatives and the ordinary derivatives in order to apply the chain rule:

Let's substitute these quantities in the formula of the chain rule:

Let's substitute  eܑ^-t by 1/e^t  we have:

dz/dt = 2 e^6t +1 / e^t⎷e^6t - 1

The same result can be obtained by substituting x(t) and y(t) into z = f(x, y) and then differentiating with respect to t:

Let''s differentiate using the chain rule for a function of a single variable:

Practice

Calculate dz/dt given the following functions:

Express the final answer in terms of t.