Calculus – Derivatives

Definition of Derivative:

The derivative of a function f at a number a, denoted by f ‘(a) is

if this limit exist.

Or can be written as

Interpretation of the Derivative as the Slope of a Tangent

The tangent line to y = f(x) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a. This means that the derivative is the slope of a curve at a given point on the curve.

Example:

Use the derivative to find the slope at any point along the following curves.

a) f(x) = 2x²

b) f(x) =

Solution:

a) f(x) = 2x²

b)

Derivative Notations

If we use the traditional notation y = f(x) to indicate that the independent variable is x and the dependent variable is y, then some common notations for the derivatives are as follows:

Videos

How to Connect Slopes and Derivatives - This video will show you how to connect slopes and derivatives.

What is a derivative?

The following video shows how to find the slope of a tangent line to a curve and gives the definition of a derivative.

The following video shows how to use the derivative to find the slope at any point along f(x) = x²

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