Final answer:
To find the equation of the tangent plane to the function z = e^(yx^5) + 3, you need to calculate the partial derivatives of z with respect to x and y, denoted as z_x and z_y, and then use them in the general tangent plane formula.
Step-by-step explanation:
To find the equation of the tangent plane to the surface given by the function z = ey·x⁵ + 3 at a specific point, we need first to calculate the partial derivatives of z with respect to x and y. Let's denote the partial derivatives of z with respect to x as z_x and with respect to y as z_y.
The general formula for the tangent plane at a point (x_0, y_0, z_0) is given by the equation: z - z_0 = z_x (x - x_0) + z_y (y - y_0).
To find z_x and z_y, we differentiate z using the chain rule:
- z_x = ey·x⁵·(5yx4)
- z_y = ey·x⁵·x⁵
Once these derivatives are calculated at the point of interest, we can substitute them into the tangent plane equation to find the exact plane. Note that a specific point (x_0, y_0, z_0) must be given to find the particular tangent plane at that point.