Slope of a Curve: Derivatives

Imagine a ball rolls along a curved track. At a particular point, how steep is the track? The curve's slope changes constantly, so we need a precise way to find the slope at exactly one point. This is the fundamental question that leads to the derivative.

• Slope of a curve is not constant
• We need the slope at a single point
• Average slope over an interval is not enough

To find the slope of a curve at a point, which approach do you think will give the most accurate result? Choose one answer.

• Think about the definition of slope
• Consider how to get a more accurate value

Adjust the second point towards the target point (x=1) on the curve y = x². Watch the secant line and its slope approach a limiting value. This visual evidence shows that the instantaneous slope exists and can be approximated.

• Observe secant slope as h decreases
• Notice the secant line approaches a specific line
• Record the limiting slope value

The evidence shows that as h → 0, the secant slope approaches a limit. This limit is the derivative: f'(a) = lim_{h→0} (f(a+h)-f(a))/h. For f(x)=x² at a=1: ((1+h)²-1)/h = (1+2h+h²-1)/h = 2+h → 2. So the slope at x=1 is exactly 2.

• Derivative is the limit of the difference quotient
• Algebraic simplification yields a formula
• At x=1, slope = 2

Not all curves have a derivative at every point. For example, the absolute value function f(x)=|x| has a sharp corner at x=0. The secant slopes from left and right approach different limits, so the derivative does not exist. This prevents overgeneralizing that all curves have slopes everywhere.

• Sharp corners cause derivative to be undefined
• Discontinuities also cause nonexistence
• Check if the function is smooth

Now use the same limit process to find the slope of f(x)=x³ at x=2. Compute derivative: f'(x) = 3x² by the power rule, so f'(2)=12. This shows the method works for power functions. More generally, the derivative gives instantaneous rate of change, like velocity from position over time.

• Compute derivative of x³
• General power rule emerges
• Interpret derivative as instantaneous slope

To find the slope of a curve at a single point, we take the limit of the average slope over smaller and smaller intervals. That limit is the derivative. It gives the exact instantaneous slope, solving our opening problem. The derivative is the foundation of calculus for understanding change.

• Derivative = instantaneous slope = limit of secant slopes
• Driving question answered
• You can now compute it for simple functions