# Tangent Line Calculator

The calculator will find the tangent line to the explicit, polar, parametric and implicit curve at the given point, with steps shown.

It can handle horizontal and vertical tangent lines as well.

The tangent line is perpendicular to the normal line.

Related calculator: Normal Line Calculator

## Your Input

**Calculate the tangent line to $$$y = x^{2}$$$ at $$$x = 1$$$.**

## Solution

We are given that $$$f{\left(x \right)} = x^{2}$$$ and $$$x_{0} = 1$$$.

Find the value of the function at the given point: $$$y_{0} = f{\left(1 \right)} = 1$$$.

The slope of the tangent line at $$$x = x_{0}$$$ is the derivative of the function, evaluated at $$$x = x_{0}$$$: $$$M{\left(x_{0} \right)} = f^{\prime }\left(x_{0}\right)$$$.

Find the derivative: $$$f^{\prime }\left(x\right) = \left(x^{2}\right)^{\prime } = 2 x$$$ (for steps, see derivative calculator).

Hence, $$$M{\left(x_{0} \right)} = f^{\prime }\left(x_{0}\right) = 2 x_{0}$$$.

Next, find the slope at the given point.

$$$m = M{\left(1 \right)} = 2$$$

Finally, the equation of the tangent line is $$$y - y_{0} = m \left(x - x_{0}\right)$$$.

Plugging the found values, we get that $$$y - 1 = 2 \left(x - 1\right)$$$.

Or, more simply: $$$y = 2 x - 1$$$.