What is a Derivative?

The slope of a curve at a single point — made visual

Speed, acceleration, stock prices, population growth — anything that changes has a rate of change. You already know how to find the slope of a straight line: rise over run. But what about a curve? The slope keeps changing at every point. The derivative answers the question: "What is the slope of this curve RIGHT HERE, at this exact point?"

The answer comes from drawing a tangent line — a line that just barely touches the curve at one point. The slope of that tangent line IS the derivative. For the parabola y = x^2, the tangent line at x = 1 has slope 2. At x = 0, the slope is 0 (the bottom of the bowl). At x = -1, the slope is -2 (the curve goes downhill).

In this lesson, you'll SEE the tangent line, watch it move along the curve, and discover the power rule — the shortcut that tells you the derivative of any power function.

Related Topics

Slope of a LineQuadratic EquationsPower Functions

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
What is a derivative?
The derivative of a function at a point is the slope of the tangent line to the curve at that point. It measures the instantaneous rate of change — how fast the function's output is changing at that exact input. If f(x) = x², then f'(x) = 2x, meaning the slope at any point x is 2x.
What is a tangent line?
A tangent line is a straight line that touches a curve at exactly one point (locally) and has the same slope as the curve at that point. It's the best linear approximation to the curve near that point. For y = x² at x = 1, the tangent line is y = 2x - 1.
What is the power rule?
The power rule states that the derivative of x^n is n \cdot x^{n-1}. For example: derivative of x² = 2x, derivative of x³ = 3x², derivative of x⁵ = 5x⁴, derivative of √x = x^(1/2) → (1/2)x^(-1/2). It works for any real exponent n.
What is the derivative of x²?
The derivative of x^2 is 2x. This means: at x = 1, the slope is 2. At x = 3, the slope is 6. At x = 0, the slope is 0 (the bottom of the parabola, where the curve is flat). At x = -2, the slope is -4 (the curve is going downhill).
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.