Functions
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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.

Graph

FAQ

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).

Related Topics

Slope of a LineQuadratic EquationsPower Functions