How to Match a Function with Its Derivative: 20 Practice Examples

Matching a function with its derivative is one of the most important visual and algebraic skills in calculus. A derivative tells you the instantaneous rate of change of a function, so the key is to compare slopes, turning points, concavity clues, and intervals where the original function increases or decreases.

TLDR: To match a function with its derivative, look first at where the original function is increasing, decreasing, or flat. The derivative is positive where the function rises, negative where it falls, and zero where the function has a horizontal tangent. Practice becomes easier when you connect graph features with derivative behavior instead of relying only on formulas.

Core Rules for Matching a Function and Its Derivative

Before working through examples, it helps to remember a few reliable principles. These rules apply whether you are matching equations, graphs, or descriptions.

• Increasing function: If f rises from left to right, then f′ is positive.
• Decreasing function: If f falls from left to right, then f′ is negative.
• Horizontal tangent: If f has a local maximum or minimum, then f′ = 0 at that point, assuming the derivative exists.
• Steepness matters: A steeper graph has a derivative with a larger magnitude.
• Concavity gives clues: If slopes are increasing, the derivative graph rises; if slopes are decreasing, the derivative graph falls.

How to Approach a Matching Problem

A dependable method is to analyze the original function in stages. First, mark the intervals where the function increases and decreases. Second, identify points where the tangent appears horizontal. Third, compare how steep the graph is in different regions. Finally, check whether the derivative should be a line, parabola, exponential curve, or another recognizable shape.

For example, if the original function is a parabola opening upward, its derivative is a line with positive slope. If the original function is cubic, its derivative is usually quadratic. If the original function is exponential, its derivative often has a similar exponential shape.

20 Practice Examples

The following examples are designed to build recognition. Each one gives a function and its correct derivative, followed by a short explanation of how to recognize the match.

1. Function: f(x) = x²
   Derivative: f′(x) = 2x
   The graph of x² is a U-shaped parabola. Its derivative is a straight line passing through the origin.
2. Function: f(x) = x³
   Derivative: f′(x) = 3x²
   A cubic curve has slopes that are never negative here except zero at x = 0; the derivative is an upward-opening parabola.
3. Function: f(x) = -x²
   Derivative: f′(x) = -2x
   Since the parabola opens downward, its slopes decrease from positive to negative, producing a line with negative slope.
4. Function: f(x) = 4x + 1
   Derivative: f′(x) = 4
   A linear function has constant slope, so its derivative is a horizontal line.
5. Function: f(x) = 7
   Derivative: f′(x) = 0
   A constant function does not change, so its derivative is zero everywhere.
6. Function: f(x) = x⁴
   Derivative: f′(x) = 4x³
   The original graph is even and very flat near zero; the derivative is an odd cubic curve.
7. Function: f(x) = √x
   Derivative: f′(x) = 1 / (2√x)
   The square root graph increases but flattens as x grows, so the derivative is positive and decreasing.
8. Function: f(x) = 1/x
   Derivative: f′(x) = -1/x²
   This function decreases on both sides of its vertical asymptote, so the derivative is always negative where defined.
9. Function: f(x) = eˣ
   Derivative: f′(x) = eˣ
   The exponential function is special because it matches its own derivative.
10. Function: f(x) = ln x
    Derivative: f′(x) = 1/x
    The natural logarithm increases slowly and flattens, so its derivative is positive and decreasing.

11. Function: f(x) = sin x
    Derivative: f′(x) = cos x
    The sine graph has maximum upward slope at x = 0, matching the cosine value of 1.
12. Function: f(x) = cos x
    Derivative: f′(x) = -sin x
    Cosine is flat at x = 0, then begins decreasing, so its derivative starts at zero and becomes negative.
13. Function: f(x) = tan x
    Derivative: f′(x) = sec² x
    Tangent is increasing wherever it is defined, so the derivative is always positive.
14. Function: f(x) = x² + 3x
    Derivative: f′(x) = 2x + 3
    A quadratic still has a linear derivative; the added linear term shifts the derivative upward.
15. Function: f(x) = 2x³ – 5x
    Derivative: f′(x) = 6x² – 5
    A cubic with turning behavior has a quadratic derivative that can be positive or negative depending on x.
16. Function: f(x) = (x – 2)²
    Derivative: f′(x) = 2x – 4
    The parabola has its minimum at x = 2, so the derivative must be zero there.
17. Function: f(x) = |x|
    Derivative: f′(x) = -1 for x < 0, 1 for x > 0, undefined at x = 0
    The sharp corner at the origin means there is no derivative at that point.
18. Function: f(x) = 1 – x³
    Derivative: f′(x) = -3x²
    The derivative is always nonpositive, showing the function is decreasing except where momentarily flat.
19. Function: f(x) = x eˣ
    Derivative: f′(x) = eˣ(x + 1)
    Product rule is needed here. The derivative changes sign at x = -1, indicating a local minimum.
20. Function: f(x) = sin(x²)
    Derivative: f′(x) = 2x cos(x²)
    Chain rule applies. The derivative combines the outer cosine behavior with the inner derivative 2x.

Common Mistakes to Avoid

• Confusing height with slope: A function can be high above the x-axis while its derivative is negative.
• Ignoring flat points: Local maxima and minima usually correspond to zeros of the derivative.
• Assuming every corner has a derivative: Sharp points, cusps, and vertical tangents require caution.
• Forgetting scale: The derivative reflects steepness, not just whether the graph moves up or down.

A Practical Checklist

When matching a function to its derivative, use this checklist in order:

1. Find where the original function is increasing or decreasing.
2. Mark where the slope appears to be zero.
3. Look for corners or discontinuities where the derivative may not exist.
4. Compare the general shape: line, parabola, cubic, exponential, or trigonometric curve.
5. Confirm with algebra if an equation is available.

With enough practice, derivative matching becomes less about memorization and more about interpretation. The derivative is not a separate mystery graph; it is a precise record of how the original function changes. If you consistently focus on slope, sign, and turning points, you will recognize correct matches with confidence.