### Multivariable Equation of the Tangent Plane to a Surface at a Point (Example 1)

In single variable calculus finding the equation of a tangent is easy. But in multivariable calculus, you'll be asked to find the equation of a tangent plane to a surface for a given point. Please see our provided example:

Find the equation of the tangent plane to the given surface at the given point, where the surface is \(z = 3{y^2} - 2{x^2} + x\) and the point is (2,-1,3).

The equation for tangent plane is given by:

\[z - {z_O} = \frac{{\partial z(2, - 1)}}{{\partial x}}(x - {x_O}) + \frac{{\partial z(2, - 1)}}{{\partial y}}(y - {y_O})\]

First find the partial derivatives and substitute in the given point:

\[\frac{{\partial z}}{{\partial x}} = - 4x + 1\]

\[\frac{{\partial z(2, - 1)}}{{\partial x}} = - 4(2) + 1\]

\[\frac{{\partial z(2, - 1)}}{{\partial x}} = - 7\]

\[\frac{{\partial z}}{{\partial y}} = 6y\]

\[\frac{{\partial z(2, - 1)}}{{\partial y}} = 6( - 1)\]

\[\frac{{\partial z(2, - 1)}}{{\partial y}} = - 6\]

And use \({z_O}\) from the initial point, as -3.

Substitute all of these into the tangent plane equation.

\[z + 3 = - 7(x - 2) + ( - 6)(y + 1)\]

\[z + 3 = - 7x + 14 - 6y - 6\]

\[z = - 7x - 6y + 5\]