Find the following derivatives.
z Subscript s and z Subscript t​, where zequals2 xy minus 2 x squared y​, xequals2 s plus 3 t​, and yequals2 s minus 3 t. Find dz/dx, dx/ds, dz/dy, dy/ds, dx/dt, dy/dt, z subscript s and z subscript t

We are given the following functions:

\[
z = 2xy - 2x^2 y
\]
\[
x = 2s + 3t
\]
\[
y = 2s - 3t
\]

### Step 1: Find \( \frac{dz}{dx} \), \( \frac{dz}{dy} \), \( \frac{dx}{ds} \), \( \frac{dy}{ds} \), \( \frac{dx}{dt} \), and \( \frac{dy}{dt} \).

#### 1.1: \( \frac{dz}{dx} \)
To differentiate \( z = 2xy - 2x^2 y \) with respect to \( x \) (keeping \( y \) as a constant):

\[
\frac{dz}{dx} = \frac{d}{dx}(2xy - 2x^2 y)
\]
Using the product rule:

\[
\frac{dz}{dx} = 2y \cdot \frac{d}{dx}(x) + 2x \cdot \frac{d}{dx}(y) - 4

Find the following derivatives: z Subscript s and z Subscript t​, where z equals 2xy - 2x^2y, x equals 2s + 3t, and y equals 2s - 3t. Find dz/dx, dx/ds, dz/dy, dy/ds, dx/dt, dy/dt, z subscript s, and z subscript t.

Learn how to find partial derivatives and apply the chain rule for multivariable functions. This problem focuses on calculating dz/dx, dz/dy, dx/ds, dy/ds, dx/dt, dy/dt, z_s, and z_t for the function z = 2xy - 2x^2y with x = 2s + 3t and y = 2s - 3t. Suitable for students at an undergraduate level studying multivariable calculus.

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