Derivatives

Derivatives are one of the central ideas in calculus because they describe how a quantity changes at an instant. On a graph, the derivative tells you the slope of the tangent line to a curve at a specific point. This connects geometry, motion, and real world change in a single concept. Students use derivatives to study speed, growth, optimization, and the shape of graphs.

The derivative is built from the slope formula for a secant line between two nearby points. As the second point moves closer to the first, the secant slope approaches the slope of the tangent line. This limiting process gives the instantaneous rate of change. In symbols, for y = f(x), the derivative at x = a is f'(a) = lim h->0 [f(a + h) - f(a)]/h.

Key Facts

Average rate of change on [a, b]: [f(b) - f(a)]/(b - a)
Derivative definition: f'(a) = lim h->0 [f(a + h) - f(a)]/h
If y = f(x), then f'(x) gives the slope of the tangent line at x
Tangent line at x = a: y - f(a) = f'(a)(x - a)
Power rule: d/dx (x^n) = n x^(n - 1)
If position is s(t), then velocity is v(t) = s'(t)

Vocabulary

Derivative: The derivative measures the instantaneous rate of change of a function and the slope of its tangent line.
Tangent line: A tangent line is the line that touches a curve at one point and has the same slope as the curve there.
Secant line: A secant line passes through two points on a curve and shows the average rate of change between them.
Limit: A limit describes the value a quantity approaches as the input gets closer to a certain point.
Instantaneous rate of change: Instantaneous rate of change is how fast a quantity is changing at one exact moment or input value.

Common Mistakes to Avoid

Using the slope between two far apart points as the derivative, which is wrong because that gives an average rate of change, not the instantaneous rate at one point.
Forgetting that the derivative is found with a limit, which is wrong because the tangent slope comes from secant slopes as the interval shrinks toward zero.
Confusing f'(a) with f(a), which is wrong because f(a) is the function value or y-coordinate, while f'(a) is the slope at that x-value.
Applying the power rule incorrectly to constants or negative exponents, which is wrong because d/dx(c) = 0 and d/dx(x^n) = n x^(n - 1) works for many real exponents when the function is defined.

Practice Questions

1 Find the average rate of change of f(x) = x^2 + 1 on the interval [1, 3].
2 For f(x) = 3x^2 - 4x + 2, find f'(x) and then find the slope of the tangent line at x = 2.
3 A graph is increasing at a point but has a horizontal tangent there. What does this tell you about the derivative at that point, and how can the function still be increasing nearby?