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Select the variables and write function with its coordinates. The tool will immediately determine the plane tangent to a point on a curve, with the steps shown.

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An online tangent plane calculator will help you efficiently determine the tangent plane at a given point on a curve. Moreover, it can accurately handle both 2 and 3 variable mathematical functions and provides a step-by-step solution. Being able to calculate a tangent plane quickly using this calculator without going through all the steps of differential calculus is a huge time saver.
Additionally, you will also get its theoretical background and will find some solved examples as a bonus.
## What is Tangent Plane?

As you know that derivative \(\frac{dy}{dx}\) of a function \(f(x)\) at a particular point represents a tangent line at that point. You can calculate tangent line to a surface using our Tangent Line Calculator. Similarly, partial derivative \(frac{∂y}{∂x}\) of function \(f(x)\) at a particular point represents a tangent plane at that point. At a point, it will contain all the tangent lines which are touching the curvature of the function under consideration at that point as shown in the figure below.
**Condition for Tangent Plane:**

The function forming the surface should be differentiable at a point so that this plane may exist there.
**Tangent Plane Equation:**

Let S be a surface defined by a differentiable function \( z = f(x,y) \) which involves 2 variables, and let \(P_o = (x_o, y_o)\) be a point in the domain of f. Then, the equation of tangent plane to S at Po is given by:
$$ z = f(x_o,y_o )+f_x(x_o,y_o )(x-x_o )+f_y(x_o,y_o )(y-y_o )$$
On similar lines, general equation of tangent plane at \(P_o=(x_o, y_o, z_o)\) to a surface S defined by a mathematical function \(z = f(x,y,z)\) which involves 3 variables is given below:
$$z = f(x_o, y_o, z_o) + f_x(x_o, y_o, z_o)(x − x_o) + f_y(x_o, y_o, z_o)(y − y_o) + f_z(x_o, y_o, z_o)(z − z_o)$$
**How to Find a Tangent Plane Equation?**

You need to follow the forthcoming steps for finding the equation of a tangent plane on a surface given by a function. This tangent plane calculator also gives a similar solution in a fraction of the time.
**Checking the Pre-Requisites:**

Make sure that you have a mathematical function of the surface and the coordinates of the point on that surface where you want to calculate the equation.
**Solving Partial Differentials:**

Partially differentiate the mathematical function of the surface under consideration. The detailed calculations can be seen from the examples shown in the next section.
**Calculating Partial Differentials at a Point:**

Calculate the value of partially differentiated function at the given points for finding tangent plane equation as shown in the upcoming examples.
**Solved Examples:**

Following examples clearly illustrate how the desired equation can be determined using the above-mentioned steps.
Our tangent plane calculator also follows the same procedure as used in these examples and you can get exactly same result in seconds.
**Example-1:**
Find the equation of the tangent plane to the surface \(z=x2+y2\) at the point \((1,2,5)\).
**Solution:**
For the function \(f(x,y) = x^2+y^2\) , we have:
$$fx(x,y) = 2x$$
$$fy(x,y) = 2y$$
So, the equation of the tangent plane at the point \((1,2,5)\) is:
$$2(1)(x−1)+2(2)(y−2)−z+5 = 0$$
$$= 2x+4y−z−5=0$$
**Example-2:**
Find the equation of the tangent plane to the surface defined by the function \(f(x,y)=sin(2x)cos(3y)\) at the point \((π/3,π/4)\).
**Solution:**
First, we will calculate \(fx(x,y)\) and \(fy(x,y)\), then we’ll calculate the required tangent plane equation using the general equation \(z=f(x_o,y_o )+fx(x_o,y_o )(x-x_o )+fy(x_o,y_o)(y-y_o)\) with \(xo = \frac{π}{3}\) and \(yo = \frac{π}{4}\):
$$f_x(x,y) = 2cos(2x)cos(3y)$$
$$f_y(x,y) = −3sin(2x)sin(3y)$$
$$f(\frac{π}{3},\frac{π}{4}) = sin(2(\frac{π}{3}))cos(3(\frac{π}{4})) = (\frac{\sqrt{3}}{2})(\frac{-\sqrt{2}}{2}) = \frac{-\sqrt{6}}{2}$$
$$f_x(\frac{π}{3},\frac{π}{4}) = 2cos(2(\frac{π}{3}))cos(3(\frac{π}{4})) = 2(\frac{-1}{2})( \frac{-√2}{2}) = \frac{\sqrt{2}}{2}$$
$$f_y(\frac{π}{3},\frac{π}{4}) = 2\sqrt{2} − 3sin(2(\frac{π}{3}))sin(3(\frac{π}{4})) = −3(3\sqrt{2})(2\sqrt{2}) = −36\sqrt{4}$$
Now, we will Substitute these values in the general equation:
$$z = f(x_o,y_o) + f_x(x_o,y_o)(x−x_o) + f_y(x_o,y_o)(y−yo_)$$
$$z = −6\sqrt{4} + 2\sqrt{2}(x − \frac{π}{3}) − 36√4 (y − \frac{π}{4})$$
$$= \frac{\sqrt{2}}{2}x − \frac{(3\sqrt{6})}{4}y − \frac{\sqrt{6}}{4} − \frac{π\sqrt{2}}{6} + \frac{3π\sqrt{6}}{16}$$
**Example-3:**
Find the tangent plane to \(x^2+ y^2 + z^2 = 30\) at the point \((1, -2, 5)\).
**Solution:**
$$f(x, y, z)=x^2 + y^2 + z^2$$
$$∇f = (2x,2y,2z)$$
$$∇f(1,−2,5) = (2,−4,10)$$
$$\text{ Solution Equation } = 2(x - 1) - 4(y + 2) + 10(z - 5)$$
**Example-4:**
Determine the tangent plane to the surface \(x2 + 2y2 + 3z2 = 36\) at the point \(P = (1, 2, 3)\)
**Solution:**
In order to use gradients, we introduce a new variable:
$$w = x^2 + 2y^2 + 3z^2$$
Our surface is then the level surface \(w = 36\). Therefore, the normal to surface is:
$$∇w = (2x, 4y, 6z)$$
At the point P we have \(∇w|P = (2, 8, 18)\). Using point normal form, the equation of the tangent plane is:
$$2(x − 1) + 8(y − 2) + 18(z − 3) = 0, \text { or equivalently } 2x + 8y + 18z = 72$$
**How to Use Tangent Plane Calculator:**

Efficient and speedy calculation equation for tangent plane is possible by this online calculator by following the forthcoming steps:
You can toggle between 2-variable calculation and 3-variable calculation by hitting the relevant tabs that are on the top of input fields.
**Inputs:**

**Outputs:**

This calculator determines the equation of the tangent plane touching the surface (formed by given mathematical function) at the coordinate points. It also provides a step-by-step solution entailing all the relevant details differentiation.
**FAQs:**

**What is the basic mathematical framework used for determining tangent plane?**

Partial differentiation is basically used to determine its equation which governs the plane. This tangent plane calculator is based on the same mathematical concept and yields accurate results in seconds.
**Does tangent plane lie in 2-D space or 3-D space?**

Tangent lines lie in 2-D space, but tangent planes are a combination of all the tangent lines touching a surface at a particular point hence, it lies in 3-D space.
**What is the difference between tangent vector and tangent plane?**

Tangent vector is a single line which barely touches the surface (determined by a mathematical function) at a point whereas, tangent plane is a combination of all the tangent vectors touching the surface at a particular point.
**What is the correlation between tangent plane and normal line?**

A tangent plane barely touches the curve surface and runs parallel to it whereas, a normal line passes through the surface and in perpendicular to it.
**Conclusion:**

Performing all these calculations manually is a very tedious process. This online tangent plane equation calculator is a handy resource which produces accurate results in no time even when dealing with 3 variable functions. The mathematical framework used in the backend calculation is exactly the same as used in the manual process.
**References:**

From Wikipedia – Tangents
From the educational blog of Openstax – Tangent Planes and Linear Approximations
From the online resources of Libretexts – Tangent Plane to a Surface

- Firstly, enter the desired mathematical function in the input field titled “Enter a Function”.
- Then simply enter the coordinates depending on the number of variables in the function.

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