Week 6: The chain rule, Directional derivatives and gradient vectors, Tangent planes and differentials

Thomas Calculus, 11th edition, pages: 996 - 1026

Sec 14.4 The Chain Rule 

The Chain Rule for functions of a single variable studied in Section 3.5 said that when w = f(x) was a differentiable function of x and x = g(t) was a differentiable function of t, w became a differentiable function of t and dw /dt could be calculated with the formula
dw / dt = dw/dx *dx/dt
For functions of two or more variables the Chain Rule has several forms. The form depends on how many variables are involved but works like the Chain Rule in Section 3.5 once we account for the presence of additional variables.
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Sec 14.5 Directional Derivative and Gradient Vectors

In the section we introduce the concept of directional derivatives and gradient vectors, including how to compute them in planes and see a couple of facts and properties pertaining to directional derivatives.

Sec 14.6 Tangent Planes and Differentials

In this section we define the tangent plane at a point on a smooth surface in space. We calculate an equation of the tangent plane from the partial derivatives of the function defining the surface. This idea is similar to the definition of the tangent line at a point on a curve in the coordinate plane for single-variable functions (Section 2.7). We then study the total differential and linearization of functions of several variables.