A. The Chain Rule: Suppose x and y are differentiable functions of t and is differentiable. Then z is differentiable with respect to t and

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Suppose that x = g(s, t), y =h(s,t) and z = f(s,t). In this case the version of the Chain Rule for computing the partial derivatives of z with respect to s and t is

Both versions of the Chain Rule may be generalized to functions of more than two variables.

PROBLEMS

1. Find where z = x²y, x = e^t, y = t²

2. Find

B. Implicit differentiation You can perform implicit differentiation for a function in one variable. If y = f(x) is defined implicitly by the equation F(x, y) = 0, then .

Verify this be finding y’ for x² + y² = 1. Begin by rewriting this equation in the form F(x, y) = 0.

Moving to functions of more than variable, if is defined implicitly by
F( x, y, z ) = 0, then .

PROBLEMS

3. Find y’ where y is defined by

4. Find the equation of the tangent plane to x² + 2y² + z² = 6 at the point
(2, -1 1).