Calculus – Derivatives
The study of differential calculus is concerned with how one quantity changes in relation to another quantity. The central concept of differential calculus is the derivative.
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
What is a derivative?
Understanding the Definition of the DerivativeThe following video shows how to use the derivative to find the slope at any point along f(x) = x2
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