Derivatives Summary

Learning Outcomes

  • Find the slope and equation of tangent lines to a function's graph.

  • Find the equation of a normal line to a function's graph.

  • Apply derivatives to differentiate functions.

Tangent Line

  • Definition: A line that touches the graph of a function at a single point.

  • Slope derived from limit: m=racf(x+h)−f(x)hm = rac{f(x+h) - f(x)}{h} as h→0h \to 0.

  • Secant line connects two points, tangent approaches as two points converge.

Normal Line

  • Definition: A line perpendicular to the tangent line at a point.

  • Relationship of slopes for perpendicular lines: m<em>1⋅m</em>2=−1m<em>1 \cdot m</em>2 = -1.

Derivative of a Function

  • Definition: The derivative f′(x)f'(x) defined as f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.

  • Differentiation: The process of finding the derivative.

  • Interpretation: The derivative represents the instantaneous rate of change and slope of the tangent line.

Example Derivative Calculation

  • Given function: f(x)=x3−4xf(x) = x^3 - 4x.

  • Find derivatives and evaluate at specific points to interpret results.