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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 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

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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