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Derivative as Slope
Limits, Tangent Lines, and Rates of Change
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The derivative of a function at a point measures its instantaneous rate of change — the slope of the tangent line at that point. It is defined as a limit: f'(x) = lim[h→0] (f(x+h) − f(x))/h. This expression is the slope of a secant line (connecting two points on the curve) as the two points get infinitely close together. When the limit exists, the function is differentiable at that point and the derivative gives the exact slope of the curve at that instant.
Derivatives have countless real-world interpretations: velocity is the derivative of position with respect to time; acceleration is the derivative of velocity; marginal cost is the derivative of total cost. The power rule (d/dx[xⁿ] = nxⁿ⁻¹), product rule, quotient rule, and chain rule form a toolkit for computing derivatives algebraically without going back to the limit definition every time. A positive derivative means the function is increasing; a negative derivative means decreasing; a zero derivative indicates a potential local maximum, minimum, or inflection point.
Key Facts
- f'(x) = lim[h→0] (f(x+h) − f(x))/h — the limit definition of the derivative
- Power rule: d/dx[xⁿ] = nxⁿ⁻¹
- Product rule: d/dx[f·g] = f'g + fg'
- Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
- f'(x) > 0: function increasing; f'(x) < 0: decreasing; f'(x) = 0: critical point
- Second derivative test: f''(x) > 0 → local min; f''(x) < 0 → local max at a critical point
Vocabulary
- Derivative
- The instantaneous rate of change of a function at a point; geometrically, the slope of the tangent line to the graph at that point.
- Tangent line
- A line that touches a curve at exactly one point (locally) and has the same slope as the curve at that point.
- Secant line
- A line connecting two points on a curve; as the points approach each other, the secant slope approaches the derivative.
- Critical point
- A point where f'(x) = 0 or f'(x) is undefined; candidates for local maxima, minima, or inflection points.
- Differentiable
- A function is differentiable at a point if the derivative exists there; requires the function to be continuous and smooth (no sharp corners or cusps).
Common Mistakes to Avoid
- Confusing the average rate of change with the instantaneous rate. Average rate = Δy/Δx over an interval; instantaneous rate = f'(x) at a single point via a limit.
- Applying the power rule to exponential functions. d/dx[eˣ] = eˣ (not x·eˣ⁻¹). The power rule applies when x is the base and the exponent is a constant, not when x is the exponent.
- Forgetting to apply the chain rule for composite functions. d/dx[sin(3x)] = cos(3x)·3, not just cos(3x). The derivative of the outer function must be multiplied by the derivative of the inner function.
- Concluding that f'(x) = 0 guarantees a maximum or minimum. f'(x) = 0 at inflection points too (e.g., f(x) = x³ at x = 0). Use the second derivative test or sign chart to classify.
Practice Questions
- 1 Find the derivative of f(x) = 3x⁴ − 5x² + 7. At what x-values is the tangent line horizontal?
- 2 Using the limit definition, find f'(x) for f(x) = x².
- 3 A particle's position is s(t) = t³ − 6t² + 9t. Find the velocity and acceleration functions and determine when the particle is at rest.