Derivatives

Definition of the Derivative

The derivative of a function quantifies the instantaneous rate of change of the function with respect to its independent variable. It is foundational in calculus for understanding how functions change and for finding slopes of tangent lines to curves.

• Formal Definition: The derivative of a function at a point is defined as:

• If this limit exists, is said to be differentiable at .

• If is differentiable at every point of an open interval , then is differentiable on .

• This formula provides the general function for finding the slope of the tangent line at any point on .

Synonyms and Interpretations of the Derivative

The derivative has several equivalent interpretations and synonyms in calculus:

• Slope of at : The value of the derivative at gives the slope of the curve at that point.

• Slope of the tangent to at : The tangent line touches the curve at and its slope is .

• Instantaneous rate of change: The derivative measures how changes with respect to at a specific instant.

• Difference quotient: The expression is called the difference quotient of .

• The derivative at is also expressed as .

Notation for Derivatives

There are several common notations for the derivative of a function :

• : "f prime"; concise, but does not name the independent variable.

• : "the derivative of with respect to d$ for derivative.

• : "the derivative of with respect to $x"; emphasizes the function's name.

• : "the derivative of at "; emphasizes differentiation as an operation performed on .

Examples: Finding Derivatives

Let us compute derivatives using the definition.

• Example 1: Find the derivative of .

• Example 2: Find the derivative function for .

• Example 3: Find the derivative function for .

Multiply numerator and denominator by :

• Example 4: Find the derivative function for .

Application: Tangent Lines

The derivative at a point provides the slope of the tangent line to the curve at that point. The equation of the tangent line to at is:

• Example: For , find the equation of the tangent line at .

Equation:

• To find the point where the slope of the tangent line equals a given value , solve and substitute into .

Calculator Practice: Derivatives and Tangent/Normal Lines

• Example: For , find the derivative at and the equation of the tangent line.

Equation:

• Example: For , find the derivative at and the equation of the normal line.

The normal line has slope (negative reciprocal). Equation:

Summary Table: Derivative Notations

Key Terms

• Derivative: The instantaneous rate of change of a function.

• Differentiable: A function is differentiable at a point if its derivative exists there.

• Tangent Line: A straight line that touches a curve at a single point and has the same slope as the curve at that point.

• Normal Line: A line perpendicular to the tangent line at a given point on the curve.

• Difference Quotient: The expression used to define the derivative.

Additional info: These notes cover the foundational concepts of derivatives, including their definition, notation, calculation using limits, and applications to tangent and normal lines. Examples are provided to illustrate the process of finding derivatives and equations of lines using derivatives.