Question

For the functions w, x, y, and z, express and using the chain rule and by expressing w directly in terms of u and v before differentiating. Then evaluate and at the point (u,v).

Answer 1

To express a function and using the chain rule, we need to find its partial derivatives with respect to each of its variables. Let's assume that w = f(u,v) and x = g(u,v). Using the chain rule and expressing w directly in terms of u and v, we can find the derivatives of and and evaluate them at a given point.

Step-by-step explanation:

To express a function and using the chain rule, we need to find its partial derivatives with respect to each of its variables. Let's assume that w = f(u,v) and x = g(u,v). Then, using the chain rule, we have:

$$\frac{{dz}}{{du}} = \frac{{dz}}{{dw}} \cdot \frac{{dw}}{{du}}$$

We can also express w directly in terms of u and v:

$$w = \sqrt{u^2 + v^2}$$

Now, differentiating w with respect to u using the chain rule:

$$\frac{{dw}}{{du}} = \frac{{dw}}{{du}} \cdot \frac{{du}}{{du}} + \frac{{dw}}{{dv}} \cdot \frac{{dv}}{{du}}$$

Finally, evaluate and at the point (u,v) by substituting the given values into the expressions obtained.