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Why is the derivative of x squared equal to 2x?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

The derivative of $x^{2}$ is $2 x$ because the derivative represents the instantaneous rate of change, or the slope of the tangent line at any point on a curve. Unlike a straight line, a curve does not have a constant slope, so we use a limiting process to find it.1

To find the derivative of $f (x) = x^{2}$ , we examine the slope formula for a small interval $h$ :
$\frac{f (x + h) - f (x)}{h} = \frac{(x + h)^{2} - x^{2}}{h}$

Expanding the numerator gives:
$\frac{x^{2} + 2 x h + h^{2} - x^{2}}{h} = \frac{2 x h + h^{2}}{h}$

As long as $h$ is not zero, we can simplify this to $2 x + h$ . The derivative is defined as the limit of this expression as $h$ approaches 0. As $h$ becomes infinitesimally small, the expression $2 x + h$ approaches $2 x$ . Thus, the slope of the parabola $x^{2}$ at any point $x$ is $2 x$ .12