Introduction to Derivatives, Basic Calculus

• Introduction to Derivatives, basic calculus

Review of Tangent Lines

• A tangent line is the line representing the limiting position of the secant line as it moves toward a specific point.
• The equation for the slope of the tangent line to the function at point is given by: slope = f(x) - f(x₀) / x - x₀

Derivative Definition

• Formalizing the slope of the tangent line involves using limits.
• For simplicity, the slope of the tangent line can be referred to as the derivative at a point.
• Derivatives can be denoted with simpler notations.

Derivative as a Function

• The derivative of a function with respect to is defined as the function that represents the slope of the original function at each point.
• The domain of the derivative function includes all values in the domain of the original function for which the limit exists.

Differentiability Explained

• A function is differentiable at a point if the limit exists at that point.
• If a function is differentiable at every point in an open interval, it is considered differentiable on that interval.
• If the limit exists for all real numbers, the function is differentiable everywhere.

Geometrical Interpretation

• When a function is differentiable at a point, it has a tangent line at that point.
• If the derivative does not exist, the secant lines from the right and left sides approach different lines.

Differentiability Analysis

• Differentiability can be viewed as the "closer" behavior of a function at a specific point.
• A function is differentiable if its magnified graph looks like a nonvertical line.

Differentiability and Continuity

• If a function is differentiable at a point, then it is continuous at that point.
• Polynomial functions are differentiable everywhere, and therefore, polynomial functions are continuous everywhere.

One-sided Derivatives

• If a function is not differentiable for some number, the secant lines on either side of the point approach different orientations for the tangent line.
• One-sided derivatives are introduced to further study the behavior of derivatives.