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 = rac{f(x+h) - f(x)}{h} as h \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: m1 \cdot m2 = -1.

Derivative of a Function

• Definition: The derivative f'(x) defined as f'(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) = x^3 - 4x.

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