Calculus/Differentiation

From Wikiversity

Jump to: navigation, search

Contents

[edit] Prerequisites

In order to understand and calculate derivatives, one must understand the following topics:

Calculus/Limits

Geometry

[edit] Introduction to Differentiation

Differentiation is the process of finding the slope of a line at any point on that line.

Generally, a slope can be found by taking the change in y-coordinates divided by the change in x-coordinates or:

\frac{(y_2 - y_1)}{(x_2 - x_1)} = \frac{\Delta y}{\Delta x} = m

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 Δx or Δy = 0

In order to do this, we have to undergo differentiation.

[edit] Definition of a Derivative

The following formula will calculate the slope of a line at any point:

f^\prime (x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

where f^\prime indicates the derivative, and h is the difference between a point h units away from x and x



Given f(x) = 3x2, to find the derivative of f(a) (where a is any x coordinate within the domain of f(x)), use the definition of derivative.

f^\prime (x) = \lim_{h \to 0} \frac{3(a+h)^2 - 3a^2}{h}



Here's an exercise to try:

Find f^\prime (2), given that f(x) = 4x2. The solution is below.





f^\prime(2) = \lim_{h \to 0} \frac{4(2+h)^2 - 12}{h} = \lim_{h \to 0} \frac{4(4+4h+h^2)-12}{h} = \lim_{h \to 0} \frac{16+16h+4h^2-12}{h} = \lim_{h \to 0} \frac{4+16h+4h^2}{h} = \lim_{h \to 0} (16 + 4h^2) = 16


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:

f(x) = 3x6 + sin2x − 12x + 5


Using the definition of derivative, your equation looks like this:

f^\prime(x) = \lim_{h \to 0} \frac {3(x+h)^6 + sin^2(x+h) - 12^{(x+h)} + 5 - 3x^6 + sin^x - 12^x + 5}{h}


Have fun solving that algebraically!


Luckily, this is where rules for derivatives come in.

[edit] Derivative Rules

Retrieved from "http://en.wikiversity.org/wiki/Calculus/Differentiation"