Question

Если z = e^(3u) + uv, u = u(x,y), v = v(x,y), то ∂z/∂x равно

Answer 1

Пошаговое объяснение:

$\frac{dz}{dx} = 3 {e}^{3u} \frac{du}{dx} + v \frac{du}{dx} + u \frac{dv}{dx} = \\ = (3 {e}^{3u} + v) \frac{du}{dx} + u \frac{dv}{dx}$

Answer 2

To find ∂z/∂x, we need to differentiate z with respect to x while treating u and v as functions of x and y. Let's calculate it step by step:

Given: z = e^(3u) + uv

First, we'll differentiate z with respect to u and v separately: ∂z/∂u = 3e^(3u) + v ∂z/∂v = u

Now, let's apply the chain rule to find the partial derivative with respect to x: ∂z/∂x = (∂z/∂u) * (∂u/∂x) + (∂z/∂v) * (∂v/∂x)

Note that u and v are functions of x and y, so we need to consider the chain rule for both variables.

Assuming u = u(x, y) and v = v(x, y), we have: ∂u/∂x = (∂u/∂x) * (∂x/∂x) + (∂u/∂y) * (∂y/∂x) ∂v/∂x = (∂v/∂x) * (∂x/∂x) + (∂v/∂y) * (∂y/∂x)

Since (∂x/∂x) = 1 and (∂y/∂x) = 0 (assuming x and y are independent variables), we simplify the above expressions: ∂u/∂x = (∂u/∂x) ∂v/∂x = (∂v/∂x)

Substituting these values back into the partial derivative of z with respect to x, we get: ∂z/∂x = (∂z/∂u) * (∂u/∂x) + (∂z/∂v) * (∂v/∂x) = (3e^(3u) + v) * (∂u/∂x) + u * (∂v/∂x)

So, ∂z/∂x is equal to (3e^(3u) + v) * (∂u/∂x) + u * (∂v/∂x).