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One independent variable.

Theorem 5.16   Let f be a function of two variables (x,y), differentiable on an open domain . Suppose that x and y are functions of a single variable t, differentiable on an open interval , and such that for every .

Then f ( x(t),y(t) ) is a function of t, differentiable on and we have:

Remark 5.17   Actually (x(t), y(t)) is a parametrization of a curve in the plane. The chain rule enables us to compute the derivative of a function f defined on a plane curve.

Example 5.18   Let z=f(x,y)=x2+y3. Then and .

Suppose now that and . Then and . It follows:

A direct computation shows that, as , the derivative is equal to .

Theorem 5.19   Let f be a function of three variables (x,y,z), differentiable on an open domain . Suppose that x, y and z are functions of a single variable t, differentiable on an open interval , and such that for every .

Then w=f ( x(t),y(t),z(t) ) is a function of t, differentiable on and we have:

Remark 5.20   Actually (x(t), y(t), z(t)) is a parametrization of a curve in the 3-dimensional space. The chain rule enables us to compute the derivative of a function f defined on a space curve.

Next: Two independent variables. Up: The chain rule. Previous: The chain rule.
Noah Dana-Picard
2001-05-30