Understanding Derivatives: What the Slope Function Tells You
Learn to read a derivative to find where functions increase, decrease, reach peaks and valleys — the practical skill of interpreting derivatives without graphing by hand.
Once you can differentiate, the real power is in reading the derivative. A profit function's derivative tells you whether increasing production will help or hurt. A drug concentration curve's derivative tells doctors when the medication is building up (positive rate) versus clearing out (negative rate). Finding where the derivative equals zero locates the maximum profit, minimum cost, or peak drug level. These are the kinds of decisions that calculus was invented to answer.
- Use the sign of \(f'(x)\) to determine where \(f\) is increasing or decreasing
- Find critical points by solving \(f'(x) = 0\) and identify local maxima and minima
- Interpret what derivative values mean in real-world contexts
Critical points where \(f'\) changes from \(+\) to \(-\) are local maxima; changes from \(-\) to \(+\) are local minima. No sign change means neither (called an inflection point).
Reading \(f'(x)\) for \(f(x) = x^3 - 3x^2 - 9x + 5\)
| Interval | Test \(x\) | \(f'(x)\) sign | \(f\) behavior |
|---|---|---|---|
| \((-\infty, -1)\) | \(x=-2\): \(f'=15\) | + | Increasing ↑ |
| \(x = -1\) | Critical point | 0 | Local Max |
| \((-1, 3)\) | \(x=0\): \(f'=-9\) | − | Decreasing ↓ |
| \(x = 3\) | Critical point | 0 | Local Min |
| \((3, \infty)\) | \(x=4\): \(f'=15\) | + | Increasing ↑ |
Find the critical points of \(f(x) = x^3 - 3x^2 - 9x + 5\).
\[ f'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x-3)(x+1) = 0 \]Critical points at \(x = -1\) and \(x = 3\).
Classify the critical points of \(f(x) = x^3 - 3x^2 - 9x + 5\) using the sign chart above.
- \(x = -1\): \(f'\) changes \(+ \to -\) → Local Maximum. Value: \(f(-1) = -1-3+9+5 = 10\).
- \(x = 3\): \(f'\) changes \(- \to +\) → Local Minimum. Value: \(f(3) = 27-27-27+5 = -22\).
Revenue \(R(q) = 120q - q^2\), Cost \(C(q) = q^2 + 20q + 500\). Find the output that maximizes profit.
\[ P(q) = R(q) - C(q) = 120q - q^2 - q^2 - 20q - 500 = -2q^2 + 100q - 500 \] \[ P'(q) = -4q + 100 = 0 \Rightarrow q = 25 \] \[ P'(20) = 20 > 0 \text{ (rising)}, \quad P'(30) = -20 < 0 \text{ (falling)} \Rightarrow q = 25 \text{ is max} \]Profit is maximized at 25 units: \(P(25) = -2(625) + 100(25) - 500 = \$750\).
- For \(f(x) = x^2 - 6x + 8\), find where \(f'(x) = 0\). Is it a max or min?
- If \(f'(x) > 0\) on \((2, 5)\), what is \(f\) doing on that interval?
- The derivative changes from − to + at \(x = 7\). What type of critical point is this?
▶ Show Answers
- \(f'(x) = 2x - 6 = 0 \Rightarrow x = 3\). Parabola opens up → local minimum.
- \(f\) is increasing on \((2, 5)\).
- Local minimum (valley).
- Assuming every critical point is a max or min: The derivative might equal zero without changing sign — that's an inflection point, not an extremum.
- Not checking both sides of a critical point: Always test a value to the left AND right to determine sign change.
- Confusing local and global extrema: A local max might not be the highest point on the entire domain.
- \(f'(x) > 0\) → increasing; \(f'(x) < 0\) → decreasing; \(f'(x) = 0\) → critical point.
- First Derivative Test: check the sign of \(f'\) on each side of a critical point to classify it.
- Setting \(f'(x) = 0\) and solving finds the input that maximizes or minimizes a quantity.
- This skill (optimization) is one of the most broadly applied techniques from calculus.
Setting a derivative to zero and solving for the optimum is the mathematical core of operations research — the discipline of optimizing logistics, inventory, staffing, and supply chains. When Amazon determines the optimal warehouse quantity to minimize total cost (holding + ordering), it is solving exactly this type of problem. Business analysts use derivative-based optimization in pricing models, marketing spend allocation, and production planning daily.
Calculator Connection
Use the Derivative Calculator to compute \(f'(x)\) and the Function Plotter to visualize both \(f(x)\) and \(f'(x)\) together — confirming where the function rises, falls, and reaches peaks or valleys.
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Apply what you've learned with these tools.
Understanding Derivatives: What the Slope Function Tells You - Quiz
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