# Tangent Plane Calculator

The calculator will try to find the tangent plane to the explicit and the implicit curve at the given point, with steps shown.

$($
,
,
$)$

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Calculate the tangent plane to $x^{2} + y^{2} + z^{2} = 14$ at $\left(x, y, z\right) = \left(1, 3, 2\right)$.

## Solution

The function can be represented in the form $F{\left(x,y,z \right)} = 0$, where $F{\left(x,y,z \right)} = x^{2} + y^{2} + z^{2} - 14$.

Find the partial derivatives.

$\frac{\partial}{\partial x} \left(F{\left(x,y,z \right)}\right) = \frac{\partial}{\partial x} \left(x^{2} + y^{2} + z^{2} - 14\right) = 2 x$ (for steps, see partial derivative calculator).

$\frac{\partial}{\partial y} \left(F{\left(x,y,z \right)}\right) = \frac{\partial}{\partial y} \left(x^{2} + y^{2} + z^{2} - 14\right) = 2 y$ (for steps, see partial derivative calculator).

$\frac{\partial}{\partial z} \left(F{\left(x,y,z \right)}\right) = \frac{\partial}{\partial z} \left(x^{2} + y^{2} + z^{2} - 14\right) = 2 z$ (for steps, see partial derivative calculator).

Evaluate the derivatives at the given point.

$\frac{\partial}{\partial x} \left(x^{2} + y^{2} + z^{2} - 14\right)|_{\left(\left(x, y, z\right) = \left(1, 3, 2\right)\right)} = \left(2 x\right)|_{\left(\left(x, y, z\right) = \left(1, 3, 2\right)\right)} = 2$

$\frac{\partial}{\partial y} \left(x^{2} + y^{2} + z^{2} - 14\right)|_{\left(\left(x, y, z\right) = \left(1, 3, 2\right)\right)} = \left(2 y\right)|_{\left(\left(x, y, z\right) = \left(1, 3, 2\right)\right)} = 6$

$\frac{\partial}{\partial z} \left(x^{2} + y^{2} + z^{2} - 14\right)|_{\left(\left(x, y, z\right) = \left(1, 3, 2\right)\right)} = \left(2 z\right)|_{\left(\left(x, y, z\right) = \left(1, 3, 2\right)\right)} = 4$

The equation of the tangent plane is $\frac{\partial}{\partial x} \left(F{\left(x,y,z \right)}\right)|_{\left(\left(x, y, z\right) = \left(x_{0}, y_{0}, z_{0}\right)\right)} \left(x - x_{0}\right) + \frac{\partial}{\partial y} \left(F{\left(x,y,z \right)}\right)|_{\left(\left(x, y, z\right) = \left(x_{0}, y_{0}, z_{0}\right)\right)} \left(y - y_{0}\right) + \frac{\partial}{\partial z} \left(F{\left(x,y,z \right)}\right)|_{\left(\left(x, y, z\right) = \left(x_{0}, y_{0}, z_{0}\right)\right)} \left(z - z_{0}\right) = 0.$

In our case, $2 \left(x - 1\right) + 6 \left(y - 3\right) + 4 \left(z - 2\right) = 0$.

This can be rewritten as $2 x + 6 y + 4 z = 28$.

Or, more simply: $z = - \frac{x + 3 y - 14}{2}$.

The equation of the tangent plane is $z = - \frac{x + 3 y - 14}{2} = - 0.5 x - 1.5 y + 7$A.