Multivariable Calculus · free preview

Partial Derivatives

Partial derivatives as slopes of vertical traces, the mechanics of differentiating one variable while the others stand still, and the notation that keeps the moving variable straight.

1. Differentiate one variable and freeze the rest

Differentiate one variable and freeze the rest

2. The mechanics on a worked example

The mechanics on a worked example

3. A partial derivative is a slope, not the whole story

A partial derivative is a slope, not the whole story

4. Partials of functions of three variables

Partials of functions of three variables

5. Freezing the right variable

Compute $f_x$ and $f_y$ for $f(x, y) = x^2 y^3$ .

6. Evaluating a partial at a point

For $f(x, y) = x^2 y^3$ , what is $f_y(1, 2)$ ?

7. A chain-rule partial

For $f(x, y) = \sin(xy)$ , what is $f_x(\pi, 1)$ ?

8. A three-variable partial

For $f(x, y, z) = x^2 z + yz^2$ , compute $f_z$ and give the term that involves $y$ , in the form like 2*y*z or y*z^2.

9. The slope of a vertical trace

For the surface $z = x^2 + y^2$ , walk from the point $(1, 1)$ in the pure $x$ direction. Give the slope of the surface along that walk, as a plain number.

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