Difference between revisions of "Derivative/Definition"

(Created page with 'Differential Calculus is a sub-field of Calculus that primarily focuses on how functions change as the input changes. In Differential Calculus we usually use Differentiation, or …')
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Differential Calculus is a sub-field of Calculus that primarily focuses on how functions change as the input changes. In Differential Calculus we usually use Differentiation, or the process of finding the derivative.  
 
Differential Calculus is a sub-field of Calculus that primarily focuses on how functions change as the input changes. In Differential Calculus we usually use Differentiation, or the process of finding the derivative.  
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Derivative represents the slope of the slope of the line tangent to a function at some point.
 
Derivative represents the slope of the slope of the line tangent to a function at some point.
  
Long method for Derivative: Let the function be <math>ax^n+bx^{n-1}+cx^{n-2}+....z=0</math>. Find the First Derivate
 
  
<math>\boxed{\text{Solution:}}</math>
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Long method for Derivative: Let the function be <math>f(x)=ax^n+bx^{n-1}+cx^{n-2}+ \cdots z=0</math>. Find the First Derivative.
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<math>\boxed{\text{Solution:}}</math>  
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If we imagine the secant line intersecting a curve at the points <math>A</math> and <math>B</math>. Then we can change this to the tangent by setting <math>B</math> on top of <math>A</math>. Let us call the horizontal or vertical distance as <math>h</math>.
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<math>\lim_{h\to0} \frac{f(x+h)-f(x)}{h}</math>
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<math>\implies lim_{h\to0} \frac{a(x+h)^n+b(x+h)^{n-1}+c(x+h)^{n-2}+ \cdots z-(ax^n+bx^{n-1}+cx^{n-2}+ \cdots z)}{h}</math>
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After canceling like terms we should have all terms contain an <math>h</math>. We can then cancel out the <math>h</math> and set <math>h=0</math>. Our end result is the first-derivative.
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The first derivative is denoted as <math>f'(x)</math>.

Revision as of 17:03, 3 March 2010

Differential Calculus is a sub-field of Calculus that primarily focuses on how functions change as the input changes. In Differential Calculus we usually use Differentiation, or the process of finding the derivative.


Derivative represents the slope of the slope of the line tangent to a function at some point.


Long method for Derivative: Let the function be $f(x)=ax^n+bx^{n-1}+cx^{n-2}+ \cdots z=0$. Find the First Derivative.

$\boxed{\text{Solution:}}$ If we imagine the secant line intersecting a curve at the points $A$ and $B$. Then we can change this to the tangent by setting $B$ on top of $A$. Let us call the horizontal or vertical distance as $h$.

$\lim_{h\to0} \frac{f(x+h)-f(x)}{h}$

$\implies lim_{h\to0} \frac{a(x+h)^n+b(x+h)^{n-1}+c(x+h)^{n-2}+ \cdots z-(ax^n+bx^{n-1}+cx^{n-2}+ \cdots z)}{h}$

After canceling like terms we should have all terms contain an $h$. We can then cancel out the $h$ and set $h=0$. Our end result is the first-derivative.

The first derivative is denoted as $f'(x)$.