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Derivative (examples)

 

 
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ider f(x) = 5:



The derivative of a constant function
Constant function

In mathematics, a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f = 4, then f is constant since f maps any value to 4....
 is zero
0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
.

ider the graph of . If the reader has an understanding of algebra
Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
 and the Cartesian coordinate system
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
, the reader should be able to independently determine that this line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 has a slope of 2 at every point. Using the above quotient (along with an understanding of the limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
, secant
Secant

Secant is a term in mathematics. It comes from the Latin secare . It can refer to:* a secant line, in geometry* the Trigonometric functions#Reciprocal functions, reciprocal to the cosine....
, and tangent) one can determine the slope at (4,5):

The derivative and slope are equivalent.

differentiation, one can find the slope of a curve.






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Example 1

Consider f(x) = 5:



The derivative of a constant function
Constant function

In mathematics, a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f = 4, then f is constant since f maps any value to 4....
 is zero
0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
.

Example 2

Consider the graph of . If the reader has an understanding of algebra
Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
 and the Cartesian coordinate system
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
, the reader should be able to independently determine that this line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 has a slope of 2 at every point. Using the above quotient (along with an understanding of the limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
, secant
Secant

Secant is a term in mathematics. It comes from the Latin secare . It can refer to:* a secant line, in geometry* the Trigonometric functions#Reciprocal functions, reciprocal to the cosine....
, and tangent) one can determine the slope at (4,5):

The derivative and slope are equivalent.

Example 3

Via differentiation, one can find the slope of a curve. Consider :

  
  
  
  
  


For any point x, the slope of the function is .

Example 4

Consider :

  
  
  
  
  
  


Example 5

The same as the previous example, but now we search the derivative of the derivative.
Consider :

Example 6

Consider