Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Derivative/Definition"
(→‎Problems)
 
(7 intermediate revisions by 4 users not shown)
Line 1: Line 1:
−
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.  
+
The '''[[derivative]]''' of a [[function]] is defined as the instantaneous rate of change of the function at a certain [[point]].  For a [[line]], this is just the [[slope]]. For more complex [[curves]], we can find the rate of change between two points on the curve easily since we can draw a line through them.
  
 +
<center>[[Image:derivative1.PNG]]</center>
  
 +
In the image above, the average rate of change between the two points is the slope of the line that goes through them: <math>\frac{f(x+h)-f(x)}h</math>.
  
−
Derivative represents the slope of the slope of the line tangent to a function at some point. We can also find critical points with the first and second derivative.
+
We can move the second point closer to the first one to find a more accurate value of the derivative.  Thus, taking the [[limit]] as <math>h</math> goes to 0 will give us the derivative of the function at <math>x</math>:
  
 +
<center>[[Image:derivative2.PNG]]</center>
  
 +
<center><math> f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}h. </math></center>
  
−
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.
+
If this limit exists, it is the derivative of <math>f</math> at <math>x</math>. If it does not exist, we say that <math>f</math> is not differentiable at <math>x</math>. This limit is called '''Fermat's difference quotient'''.
  
−
<math>\boxed{\text{Solution:}}</math>  
+
== Examples ==
−
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>.
+
We can apply the Fermat's difference quotient to a polynomial of the form <math>f(x)=ax^n+bx^{n-1}+cx^{n-2}+ \cdots + z=0</math> in order to find its derivative. 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>.
  
 
<math>\lim_{h\to0} \frac{f(x+h)-f(x)}{h}</math>
 
<math>\lim_{h\to0} \frac{f(x+h)-f(x)}{h}</math>
Line 65: Line 69:
  
  
−
== Problems for Part I ==
+
=== Problems ===
  
 
<math>\boxed{\text{Problem 1}}</math>: Find the first derivative of <math>f(x)</math>, where <math>f(x)=2x^2-15x+7</math>.  
 
<math>\boxed{\text{Problem 1}}</math>: Find the first derivative of <math>f(x)</math>, where <math>f(x)=2x^2-15x+7</math>.  
Line 78: Line 82:
  
  
−
<math>\boxed{\text{Problem 2}}</math>: Find the equation of the line tangent to the function <math>f(x)=3x^3-5x^2+12</math> at <math>(-1,14)</math>.
+
<math>\boxed{\text{Problem 2}}</math>: Find the equation of the line tangent to the function <math>f(x)=3x^3-5x^2+12</math> at the point <math>(-1, 4)</math>.
  
  
Line 93: Line 97:
 
<math>y=19x+b</math>.
 
<math>y=19x+b</math>.
  
−
Plugging <math>(-1,14)</math> in gives:
+
Plugging <math>(-1,4)</math> in gives <math>4=-19+b</math> and so <math>b=23</math>.
  
−
<math>14=-19+b</math>
+
Thus, the equation of the line is <math>y=19x+23</math>. Alternatively, one could use [[point-slope form]] for the line; after determining that the slope is <math>19</math>, as above, this allows one to immediately write down the equation <math>y - 4 = 19(x + 1)</math> of the line.
−
 
 
−
<math>\implies b=30</math>.
 
−
 
 
−
 
 
−
 
 
−
<math>\therefore</math> The equation of the line is <math>y=19x+33</math>.
 
  
 +
(Notice that it is implicit in the question that the point <math>(-1, 4)</math> lies on the graph of <math>y = f(x)</math>; it's easy to check that this is actually the case.)
  
  
Line 127: Line 126:
  
  
−
<math>\therefore</math> The nth derivate of <math>f(x)</math> is <math>n!</math>.
+
<math>\therefore</math> The nth derivative of <math>f(x)</math> is <math>n!</math>.
 +
 
 +
== See also ==
 +
* [[Calculus]]
 +
* [[Derivative]]
 +
 
 +
[[Category:Calculus]]

Latest revision as of 15:21, 3 January 2012

The derivative of a function is defined as the instantaneous rate of change of the function at a certain point. For a line, this is just the slope. For more complex curves, we can find the rate of change between two points on the curve easily since we can draw a line through them.

Derivative1.PNG

In the image above, the average rate of change between the two points is the slope of the line that goes through them: $\frac{f(x+h)-f(x)}h$.

We can move the second point closer to the first one to find a more accurate value of the derivative. Thus, taking the limit as $h$ goes to 0 will give us the derivative of the function at $x$:

Derivative2.PNG
$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}h.$

If this limit exists, it is the derivative of $f$ at $x$. If it does not exist, we say that $f$ is not differentiable at $x$. This limit is called Fermat's difference quotient.

Examples

We can apply the Fermat's difference quotient to a polynomial of the form $f(x)=ax^n+bx^{n-1}+cx^{n-2}+ \cdots + z=0$ in order to find its derivative. 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 $f'(x)$.


This would be some tedious work so instead there is a much nicer way to find the derivative.

Let $f(x)=3x^n$. Let $g(x)=x^t+x^{n-1}+5x^{3}$

1. Find $f'(x)$.

Any function like this is: $f'(x)=3 \cdot n \cdot x^{n-1}=3n \cdot x^{n-1}$


2. Find $g'(x)$.

Breaking apart on what we used above.

$g'(x)=t \cdot x^{t-1}+(n-1) \cdot x^{n-2}+ 5 \cdot 3 \cdot x^2$

$g'(x)=t \cdot x^{t-1}+(n-1) \cdot x^{n-2}+15x^2$


Let $f(x)=-147$. Find $f'(x)$.


If the function $f(x)$ is a constant then its derivative will always be $0$.


Notation: $f'(x)$ denotes the first derivative for $f(x)$. The symbol for the second derivative is just $f''(x)$. For the third derivative it is just $f'''(x)$. Derivatives are also written as $\frac{d}{dx} f(x)$. Or if for the nth derivative they are written as $\frac{d^n}{dx^n} f(x)$.



Maximum and Minimum: We can use the first derivative to determine the maximum and the minimum points of a graph.

If $f'(x)=6x^2-24$. Then the maximum and the minimum occur when:

$6x^2-24=$, $x=2$ or $x=-2$. We can plug each back in to the original $f(x)$ if it was given, and the one with the higher y-coordinate is the maximum, while the smaller y-coordinate gives the minimum.


Below are problems for Part I. In Part II(see link below) we will begin to actually "start" the calculus with this.


Problems

$\boxed{\text{Problem 1}}$: Find the first derivative of $f(x)$, where $f(x)=2x^2-15x+7$.


$\boxed{\text{Solution 1}}$:

$f'(x)=2 \cdot 2 \cdot x^1-15 \cdot 1 \cdot x^0+0$

$f'(x)=4x-15$.


$\boxed{\text{Problem 2}}$: Find the equation of the line tangent to the function $f(x)=3x^3-5x^2+12$ at the point $(-1, 4)$.


$\boxed{\text{Solution 2}}$:

We will take the first derivative to determine the slope of the tangent line.

$f'(x)=9x^2-10x$. If this is the slope of the tangent point then we can just plug $-1$ into the $x$ coordinate to find the actual slope.

$f'(x)=9+10=19$. The slope of the line is $19$.

Let the equation be:

$y=19x+b$.

Plugging $(-1,4)$ in gives $4=-19+b$ and so $b=23$.

Thus, the equation of the line is $y=19x+23$. Alternatively, one could use point-slope form for the line; after determining that the slope is $19$, as above, this allows one to immediately write down the equation $y - 4 = 19(x + 1)$ of the line.

(Notice that it is implicit in the question that the point $(-1, 4)$ lies on the graph of $y = f(x)$; it's easy to check that this is actually the case.)


$\boxed{\text{Problem 3}}$: Find the nth derivative of $f(x)=x^n$


$\boxed{\text{Solution 3}}$:


$\frac{d}{dx} f(x)=nx^{n-1}$


$\frac{d^2}{dx^2} f(x)=n(n-1) x^{n-2}$


$\vdots$


$\frac{d^{n}}{dx^{n}} f(x)=n(n-1)(n-2) \cdots 1$


$\frac{d^{n}}{dx^{n}} f(x)=n!$


$\therefore$ The nth derivative of $f(x)$ is $n!$.

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Derivative/Definition&oldid=44131"