Question

Partial derivatives and coordinate conversions Given the function f(x,y,z)=e−⁽ˣ² ⁺ʸ² ⁺ᶻ²⁾, compute the partial derivatives ∂f/∂x​ , ∂f/∂y , ∂f∂z

Answer 1

The partial derivatives of the function f(x,y,z) = e^(-x² + y² + z²) with respect to x, y, and z are -2x * e^(-x² + y² + z²), -2y * e^(-x² + y² + z²), and -2z * e^(-x² + y² + z²), respectively.

Step-by-step explanation:

The student has asked to compute the partial derivatives of the function f(x,y,z) = e−(x² + y² + z²). To find these, we will differentiate the function with respect to each variable while treating the other variables as constants.

Partial Derivatives of f(x,y,z)

The partial derivative of f with respect to x, denoted as ∂f/∂x, is found by differentiating f with respect to x and treating y and z as constants. Applying the chain rule, we get:

∂f/∂x = -2x * e−(x² + y² + z²)

Similarly, the partial derivative with respect to y, ∂f/∂y, is:

∂f/∂y = -2y * e−(x² + y² + z²)

And the partial derivative with respect to z, ∂f/∂z, is:

∂f/∂z = -2z * e−(x² + y² + z²)