Calculus/Differentiation
Prerequisites[edit | edit source]
In order to understand and calculate derivatives, one must understand the following topics:
Introduction to Differentiation[edit | edit source]
Differentiation is the process of finding the unique slope of a line at any point on that line, should such a slope exist. As we shall show, there are commonplace functions that fail this requirement at some point.
Generally, a slope can be found by taking the change in y-coordinates divided by the change in x-coordinates or:
where m is the slope.
Now what happens when we attempt to take the slope of a point on a line, or how fast the y's are changing in respect to the x's at a single point? The formula fails when or
In order to do this, we have to undergo differentiation.
Definition of a Derivative[edit | edit source]
The following formula will calculate the slope of a line at any point:
where indicates the derivative, and h is the difference between a point h units away from x and x
Given , to find the derivative of f(a) (where a is any x coordinate within the domain of f(x)), use the definition of derivative.
Here's an exercise to try:
Find , given that . The solution is below.
Therefore, the slope of 4x^2 at x=2 is 16.
This is a somewhat tedious process when bigger functions are involved. Take for example:
Using the definition of derivative, your equation looks like this:
Have fun solving that algebraically!
Luckily, this is where rules for derivatives come in.
Where Derivation Fails[edit | edit source]
Two examples of commonplace functions either have ambiguous derivatives or none at some point. The absolute value function has an ambiguous derivative at x=0 For |x|, |x| = -x and the derivative of |x| is -1 if x < 0. |x| = x and the derivative of |x| is 1 if x > 0. But if x=0, then |x|, x, and -x are all zero. No matter how close a number is to zero, the derivative of |x| is 1 for x > 0 but -1 for any x less than zero. Because one cannot determine whether the derivative of |x| is -1, 1, or something else at x=0, one has an ambiguous derivative at x=0... and thus none.
Another function that most off us know well is 1/x. For f(x) = 1/x
implies
No derivative of 1/x can exist at x=0.
Derivative Rules[edit | edit source]
In the following table, f(x) and g(x) are functions whereas "a" is a constant number. f'(x) and g'(c) are the derivatives of the functions f(x) and g(x), respectively.
Rule Name | Original Function | Derivative |
---|---|---|
Chain Rule | f(g(x)) | f'(g(x))•g'(x) |
constant function derivative | a | 0 |
Addition Rule | f(x)+g(x) | f'(x)+g'(x) |
??? | x | 1 |
Scalar Rule | a•f(x) | a•f'(x) |
Power Rule | (f(x))^{a} | a•(f(x))^{a-1}ˆ•f'(x) |
Product Rule | f(x)•g(x) | f(x)•g'(x) + g(x)•f'(x) |
Quotient Rule | f(x)/g(x) | (g(x)•f'(x) - f(x)•g'(x)) / (g(x))^{2} |
derivative of trigonometric function | sin(x) | cos(x) |
derivative of trigonometric function | cos(x) | -sin(x) |
derivative of trigonometric function | tan(x) | sec^{2}(x) |
derivative of trigonometric function | ln(x) | 1/x |
derivative of trigonometric function | e^{x} | e^{x} |