The derivative is a fundamental concept in calculus. It represents the rate at which a quantity changes in relation to another quantity. In simple terms, it tells us how a function behaves as its input changes.
Basic Definition
The derivative of a function ( f(x) ) with respect to ( x ) is denoted as ( f'(x) ) or ( \frac{df}{dx} ). It is defined as the limit of the difference quotient as the change in ( x ) approaches zero:
[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} ]
Types of Derivatives
- Power Rule: The derivative of ( x^n ) is ( nx^{n-1} ).
- Product Rule: The derivative of a product of two functions ( f(x) ) and ( g(x) ) is ( f'(x)g(x) + f(x)g'(x) ).
- Quotient Rule: The derivative of a quotient of two functions ( f(x) ) and ( g(x) ) is ( \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} ).
- Chain Rule: The derivative of a composite function ( f(g(x)) ) is ( f'(g(x))g'(x) ).
Applications
Derivatives have a wide range of applications in various fields, including physics, engineering, economics, and more. Here are a few examples:
- Physics: Derivatives are used to determine velocity, acceleration, and other physical quantities.
- Engineering: They are used to design and analyze structures, machines, and systems.
- Economics: Derivatives are used to model market behavior and predict future trends.
For more information on derivatives and their applications, you can visit our Calculus Basics page.
Example
Let's find the derivative of the function ( f(x) = x^2 ).
Using the power rule, we have:
[ f'(x) = 2x ]
This means that the derivative of ( x^2 ) is ( 2x ).
Note: The graph above shows the derivative of ( f(x) = x^2 ) with respect to ( x ).
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