Difficulty: Introductory | Prerequisites: Limits (Ch. 1 Limits notes), algebra (expanding polynomials, simplifying fractions)
The derivative is what calculus is fundamentally about: measuring how fast something is changing at a specific instant. This section builds directly on limits, because the derivative is defined as a limit. You will learn to compute derivatives from the definition (the "long way") before learning shortcut rules. If the limit techniques from the first part of Chapter 1 are not solid, review those first, because every derivative calculation here uses them.
The derivative f′(x) tells you the slope of the curve at any given point. You find it by taking the limit of the difference quotient [f(x+h) − f(x)] / h as h goes to zero. Constants have a derivative of zero, and the power rule (the derivative of xⁿ is nxⁿ⁻¹) is the shortcut you will use most. To find a tangent line, you need the derivative at the point for the slope and the point itself for the equation.
Tags: derivative, differentiation, tangent line, slope, secant, dy/dx, power rule, rate of change
Derivative
The instantaneous rate of change of a function at a point. Formally, f′(x) = lim (h→0) [f(x+h) − f(x)] / h.
Think of it as: how steep the curve is at one precise spot.
Difference quotient
The expression [f(x+h) − f(x)] / h before you take the limit. It measures the average rate of change between two points on the curve.
In simple terms, it is the slope formula for a line through two nearby points on the graph.
Secant line
A straight line that passes through two points on a curve. The difference quotient gives the slope of this line. As h shrinks toward zero, the secant line becomes the tangent line.
Think of it as: the approximation that gets better as the two points get closer.
Tangent line
A straight line that touches a curve at exactly one point and has the same slope as the curve at that point. Its slope equals the derivative f′(x₀) at the point of tangency.
In simple terms, the line that just grazes the curve without cutting through it.
dy/dx
Leibniz notation for the derivative. Read as "the derivative of y with respect to x." Equivalent to f′(x), y′, or Δy/Δx in the limit.
Think of it as: a different way of writing f′(x), used especially when the function is written as y = something.
Power rule
The shortcut rule stating that the derivative of xⁿ is nxⁿ⁻¹. Derived from the limit definition but far quicker to apply.
In simple terms, bring the exponent down as a multiplier and reduce the exponent by one.
Tags: limit definition of derivative, derivative examples, derivative of constant, derivative of 1/x, derivative of x cubed, tangent line equation
Every derivative in this section follows the same four-step process:
Write out f′(x) = lim (h→0) [f(x+h) − f(x)] / h.
Expand f(x+h) and subtract f(x) in the numerator.
Simplify until you can cancel h from the numerator and denominator.
Let h = 0 in whatever remains.
f(x+h) = 100, f(x) = 100.
Numerator: 100 − 100 = 0.
0 / h = 0 for any h.
lim (h→0) 0 = 0.
Result: f′(x) = 0.
Takeaway: the graph of a constant is a flat line. No slope anywhere.
Numerator: 1/(x+h) − 1/x = [x − (x+h)] / [x(x+h)] = −h / [x(x+h)].
Divide by h: −h / [hx(x+h)] = −1 / [x(x+h)].
Let h → 0: −1 / [x(x+0)] = −1/x².
Result: f′(x) = −1/x².
The negative sign confirms that 1/x is always decreasing for positive x.
Expand (x+h)³ = x³ + 3x²h + 3xh² + h³.
Subtract x³: numerator = 3x²h + 3xh² + h³.
Factor out h: h(3x² + 3xh + h²).
Cancel h: 3x² + 3xh + h².
Let h → 0: 3x² + 0 + 0 = 3x².
Result: f′(x) = 3x².
This matches the power rule: n = 3, so 3x³⁻¹ = 3x².
Expand (x+h)² + c = x² + 2xh + h² + c.
Subtract (x² + c): numerator = 2xh + h².
Factor out h: h(2x + h).
Cancel h: 2x + h.
Let h → 0: 2x.
Result: dy/dx = 2x.
The constant c vanishes entirely. Adding a constant shifts the curve up or down but does not change its slope.
The tangent line at x = x₀ requires two pieces of information:
The point: (x₀, y₀) where y₀ = f(x₀).
The slope: m = f′(x₀).
Plug both into the point-slope form: y − y₀ = m(x − x₀).
Worked example: tangent to y = √x at x = 4.
Point: y₀ = √4 = 2, so (4, 2).
Derivative of √x using the conjugate method: f′(x) = 1 / (2√x).
Slope at x = 4: f′(4) = 1 / (2 × 2) = 1/4.
Equation: y − 2 = (1/4)(x − 4), which simplifies to y = (1/4)x + 1.
Tags: derivative formula, power rule formula, tangent line formula, point-slope form
Limit definition of the derivative: f′(x) = lim (h→0) [f(x+h) − f(x)] / h
Power rule (shortcut): If f(x) = xⁿ, then f′(x) = nxⁿ⁻¹
Constant rule: If f(x) = c (any constant), then f′(x) = 0
Tangent line equation (point-slope form): y − y₀ = f′(x₀) × (x − x₀)
Derivative of √x: f′(x) = 1 / (2√x)
Derivatives measure rates of change, which is precisely the language of business. Marginal cost is the derivative of the total cost function. Marginal revenue is the derivative of the total revenue function. Whenever someone says "how fast is this growing right now," they are asking for a derivative. The tangent line gives you a linear approximation of the function near a point, which is how forecasters estimate short-term trends from curved data.
Students often confuse the slope of the secant line with the slope of the tangent line. The secant connects two points; the tangent touches at one point. The derivative is the tangent slope, not the secant slope.
Students sometimes forget to expand (x+h)ⁿ properly. For x³, you must use the full binomial expansion: x³ + 3x²h + 3xh² + h³. Skipping terms leads to a wrong derivative.
Students often leave h in the final answer. After cancelling h and taking the limit, there should be no h left. If h appears in your result, something went wrong in the simplification.
Students sometimes think the derivative of x² + c is 2x + 1 because "the constant becomes 1." Constants do not become 1. Their derivative is 0, and they disappear entirely.
⚠️ Expect to compute at least one derivative from the limit definition (the long way). Even after you learn the power rule, the exam tests whether you understand where it comes from.
⚠️ The tangent line problem is a near-certainty on the exam: find the point, find the slope via the derivative, write the equation in point-slope form.
⚠️ Know that the derivative of a constant is zero. This is a common conceptual question.
⚠️ Be able to state the difference between f′(x) and f(x) in words. The derivative tells you the rate of change (slope), not the value of the function.
True or false: The derivative of any constant is 1.
False. The derivative of any constant is 0.
Fill in the blank: The power rule says the derivative of xⁿ is ______.
nxⁿ⁻¹.
True or false: To find a tangent line, you only need the derivative.
False. You also need the point (x₀, y₀) on the curve.
Fill in the blank: f′(x) = lim (h→0) [f(x+h) − f(x)] / ______.
h.
True or false: If f(x) = x² + 7, then f′(x) = 2x + 7.
False. The constant 7 drops out. f′(x) = 2x.
Q: Use the limit definition to find f′(x) for f(x) = x².
A: f′(x) = lim (h→0) [(x+h)² − x²] / h = lim (h→0) [2xh + h²] / h = lim (h→0) (2x + h) = 2x.
Q: Find the derivative of f(x) = 5 using the limit definition.
A: f′(x) = lim (h→0) [5 − 5] / h = lim (h→0) 0/h = 0.
Q: Find the equation of the tangent line to y = x³ at x = 2.
A: Point: y₀ = 2³ = 8, so (2, 8). Derivative: f′(x) = 3x², so f′(2) = 3(4) = 12. Tangent line: y − 8 = 12(x − 2), which gives y = 12x − 16.
Q: Why does the constant c disappear when you differentiate y = x² + c?
A: When you expand (x+h)² + c and subtract (x² + c), the c terms cancel: x² + 2xh + h² + c − x² − c = 2xh + h². The constant shifts the graph vertically but does not change the slope at any point.
Q: What is the difference between a secant line and a tangent line?
A: A secant line passes through two points on a curve and measures the average rate of change between them. A tangent line touches the curve at one point and measures the instantaneous rate of change there. The tangent is the limit of the secant as the two points merge.
This connects back to limits: every derivative is a limit, and the techniques for resolving 0/0 (factoring, conjugate multiplication) are the same ones you use to simplify the difference quotient. If limit algebra is shaky, derivatives will be harder than they need to be.
Looking ahead, the power rule and constant rule are the first of several shortcut rules (product rule, quotient rule, chain rule) that will let you skip the limit definition for more complex functions. The tangent line concept extends into linear approximation and differentials later in the course.
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